Higher engineering-mathematics-b-s-grewal-companion-text

SOLUTION

1. Scilab Textbook Companion for
Higher Engineering Mathematics
by B. S. Grewal
1
Created by
Karan Arora and Kush Garg
B.Tech. (pursuing)
Civil Engineering
Indian Institute of Technology Roorkee
College Teacher
Self
Cross-Checked by
Santosh Kumar, IIT Bombay
August 10, 2013
1
Funded by a grant from the National Mission on Education through ICT,
http://spoken-tutorial.org/NMEICT-Intro. This Textbook Companion and Scilab
codes written in it can be downloaded from the "Textbook Companion Project"
section at the website http://scilab.in

2. Book Description
Title:Higher Engineering Mathematics
Author:B. S. Grewal
Publisher:Khanna Publishers, New Delhi
Edition:40
Year:2007
ISBN:8174091955
1

3. Scilab numbering policy used in this document and the relation to the
above book.
ExaExample (Solved example)
EqnEquation (Particular equation of the above book)
APAppendix to Example(Scilab Code that is an Appednix to a particular
Example of the above book)
For example, Exa 3.51 means solved example 3.51 of this book. Sec 2.3 means
a scilab code whose theory is explained in Section 2.3 of the book.
2

4. Contents
List of Scilab Codes
1 Solution of equation and curve tting
2 Determinants and Matrices
4 Dierentiation and Applications
5 Partial Dierentiation And Its Applications
6 Integration and its Applications
9 Innite Series
10 Fourier Series
13 Linear Dierential Equations
21 Laplace Transform
22 Integral Transform
23 Statistical Methods
24 Numerical Methods
26 Dierence Equations and Z Transform
27 Numerical Solution of Ordinary Dierential Equations
3

5. 28 Numerical Solution of Partial Dierential Equations
34 Probability and Distributions
35 Sampling and Inference
4

6. List of Scilab Codes
Exa 1.1 nding the roots of quadratic equations
Exa 1.2 nding the roots of equation containing one variable
Exa 1.3 nding the roots of equation containing one variable
Exa 1.6 nding the roots of equation containing one variable
Exa 1.7 nding the roots of equation containing one variable
Exa 1.11 forming an equation with known roots
Exa 1.12 forming an equation under restricted conditions
Exa 1.13 nding the roots of equation containing one variable
Exa 1.14 nding the roots of equation containing one variable
Exa 1.15 nding the roots of equation containing one variable
Exa 1.16 nding the roots of equation containing one variable
Exa 1.17 nding the roots of equation containing one variable
Exa 1.18 Finding the roots of equation containing one variable
Exa 1.19 Finding the roots of equation containing one variable
Exa 1.20 Finding the roots of equation containing one variable
Exa 1.21 Finding the roots of equation containing one variable
Exa 1.22 Finding the roots of equation containing one variable
Exa 1.23 Finding the solution of equation by drawing graphs
Exa 1.24 Finding the solution of equation by drawing graphs
Exa 1.25 Finding the solution of equation by drawing graphs
Exa 2.1 Calculating Determinant
Exa 2.2 Calculating Determinant
Exa 2.3 Calculating Determinant
Exa 2.4 Calculating Determinant
Exa 5.8 Partial derivative of given function
Exa 2.16 product of two matrices
Exa 2.17 Product of two matrices
Exa 2.18 Product and inverse of matrices
5

7. Exa 2.19 Solving equation of matrices
Exa 2.20 Nth power of a given matrix
Exa 2.23 Inverse of matrix
Exa 2.24.1 Rank of a matrix
Exa 2.24.2 Rank of a matrix
Exa 2.25 Inverse of matrix
Exa 2.26 eigen values vectors rank of matrix
Exa 2.28 Inverse of a matrix
Exa 2.31 Solving equation using matrices
Exa 2.32 Solving equation using matrices
Exa 2.34.1 predicting nature of equation using rank of matrix
Exa 2.34.2 predicting nature of equation using rank of matrix
Exa 2.38 Inverse of a matrix
Exa 2.39 Transpose and product of matrices
Exa 2.42 eigen values and vectors of given matrix
Exa 2.43 eigen values and vectors of given matrix
Exa 2.44 eigen values and vectors of given matrix
Exa 2.45 eigen values and characteristic equation
Exa 2.46 eigen values and characteristic equation
Exa 2.47 eigen values and characteristic equation
Exa 2.48 eigen values and vectors of given matrix
Exa 2.49 eigen values and vectors of given matrix
Exa 2.50 eigen values and vectors of given matrix
Exa 2.51 eigen values and vectors of given matrix
Exa 2.52 Hermitian matrix
Exa 2.53 tranpose and inverse of complex matrix
Exa 2.54 Unitary matrix
Exa 4.4.1 nding nth derivative
Exa 4.5 nding nth derivative
Exa 4.6 nding nth derivative
Exa 4.7 nding nth derivative
Exa 4.8 proving the given dierential equation
Exa 4.9 proving the given dierential equation
Exa 4.10 proving the given dierential equation
Exa 4.11 verify roles theorem
Exa 4.16 expansion using maclaurins series
Exa 4.17 expanding function as fourier series of sine term
Exa 4.18 expansion using maclaurins series
6

8. Exa 4.19 expansion using maclaurins series
Exa 4.20 expansion using taylors series
Exa 4.21 taylor series
Exa 4.22 evaluating limit
Exa 4.32 tangent to curve
Exa 4.34 nding equation of normal
Exa 4.35 nding angle of intersection of curve
Exa 4.37 prove given tangent statement
Exa 4.39 nding angle of intersection of curve
Exa 4.41 nding pedal equation of parabola
Exa 4.43 nding radius of curvature of cycloid
Exa 4.46 radius of curvature of cardoid
Exa 4.47 cordinates of centre of curvature
Exa 4.48 proof statement cycloid
Exa 4.52 maxima and minima
Exa 4.61 nding the asymptotes of curve
Exa 5.5 Partial derivative of given function
Exa 5.14 Partial derivative of given function
Exa 5.25.1 Partial derivative of given function
Exa 5.25.2 Partial derivative of given function
Exa 5.25.3 Partial derivative of given function
Exa 5.26 Partial derivative of given function
Exa 5.30 Partial derivative of given function
Exa 6.1.1 indenite integral
Exa 6.1.2 indenite integral
Exa 6.2.1 denite integral
Exa 6.2.2 Denite Integration of a function
Exa 4.2.3 denite integral
Exa 6.2.3 denite integral
Exa 6.4.1 denite integral
Exa 4.4.2 denite integral
Exa 6.5 denite integral
Exa 6.6.1 reducing indenite integral to simpler form
Exa 6.7.1 Indenite Integration of a function
Exa 6.8 Getting the manual input of a variable and integration
Exa 6.9.1 Denite Integration of a function
Exa 6.9.2 Denite Integration of a function
Exa 6.10 denite integral
7

9. Exa 6.12 Denite Integration of a function
Exa 6.13 sum of innite series
Exa 6.14 nding the limit of the function
Exa 6.15 Denite Integration of a function
Exa 6.16 Denite Integration of a function
Exa 6.24 Calculating the area under two curves
Exa 9.1 to nd the limit at innity
Exa 9.1.3 to nd the limit at innity
Exa 9.2.1 to nd the sum of series upto innity
Exa 9.2.2 to check for the type of series
Exa 9.5.1 to check the type of innite series
Exa 9.5.2 to check the type of innite series
Exa 9.7.1 to check the type of innite series
Exa 9.7.3 to check the type of innite series
Exa 9.8.1 to nd the sum of series upto innity
Exa 9.8.2 to nd the limit at innity
Exa 9.10.1 to nd the limit at innity
Exa 9.10.2 to nd the limit at innity
Exa 9.11.1 to nd the limit at innity
Exa 9.11.2 to nd the limit at innity
Exa 10.1 nding fourier series of given function
Exa 10.2 nding fourier series of given function
Exa 10.3 nding fourier series of given function
Exa 10.4 nding fourier series of given function
Exa 10.5 nding fourier series of given function in interval minus
pi to pi
Exa 10.6 nding fourier series of given function in interval minus
l to l
Exa 10.7 nding fourier series of given function in interval minus
pi to pi
Exa 10.8 nding fourier series of given function in interval minus
pi to pi
Exa 10.9 nding half range sine series of given function
Exa 10.10 nding half range cosine series of given function
Exa 10.11 expanding function as fourier series of sine term
Exa 10.12 nding fourier series of given function
Exa 10.13 nding complex form of fourier series
Exa 10.14 practical harmonic analysis
8

10. Exa 10.15 practical harmonic analysis
Exa 10.16 practical harmonic analysis
Exa 10.17 practical harmonic analysis
Exa 13.1 solvinf linear dierential equation
Exa 13.2 solving linear dierential equation
Exa 13.3 solving linear dierential equation
Exa 13.4 solving linear dierential equation
Exa 13.5 nding particular integral
Exa 13.6 nding particular integral
Exa 13.7 nding particular integral
Exa 13.8 nding particular integral
Exa 13.9 nding particular integral
Exa 13.10 nding particular integral
Exa 13.11 solving the given linear equation
Exa 13.12 solving the given linear equation
Exa 13.13 solving the given linear equation
Exa 13.14 solving the given linear equation
Exa 21.1.1 nding laplace transform
Exa 21.1.2 nding laplace transform
Exa 21.1.3 nding laplace transform
Exa 21.2.1 nding laplace transform
Exa 21.2.2 nding laplace transform
Exa 21.2.3 nding laplace transform
Exa 21.4.1 nding laplace transform
Exa 21.4.2 nding laplace transform
Exa 21.5 nding laplace transform
Exa 21.7 nding laplace transform
Exa 21.8.1 nding laplace transform
Exa 21.8.2 nding laplace transform
Exa 21.8.3 nding laplace transform
Exa 21.8.4 nding laplace transform
Exa 21.9.1 nding laplace transform
Exa 21.9.2 nding laplace transform
Exa 21.10.1nding laplace transform
Exa 21.10.3nding laplace transform
Exa 21.11.1nding inverse laplace transform
Exa 21.11.2nding inverse laplace transform
Exa 21.12.1nding inverse laplace transform
9

11. Exa 21.12.3nding inverse laplace transform
Exa 21.13.1nding inverse laplace transform
Exa 21.13.2nding inverse laplace transform
Exa 21.14.1nding inverse laplace transform
Exa 21.14.2nding inverse laplace transform
Exa 21.15.1nding inverse laplace transform
Exa 21.15.2nding inverse laplace transform
Exa 21.16.1nding inverse laplace transform
Exa 21.16.2nding inverse laplace transform
Exa 21.16.3nding inverse laplace transform
Exa 21.17.1nding inverse laplace transform
Exa 21.17.2nding inverse laplace transform
Exa 21.19.1nding inverse laplace transform
Exa 21.19.2nding inverse laplace transform
Exa 21.28.1nding laplace transform
Exa 21.28.2nding laplace transform
Exa 21.34 nding laplace transform
Exa 22.1 nding fourier sine integral
Exa 22.2 nding fourier transform
Exa 22.3 nding fourier transform
Exa 22.4 nding fourier sine transform
Exa 22.5 nding fourier cosine transform
Exa 22.6 nding fourier sine transform
Exa 23.1 Calculating cumulative frequencies of given using itera-
tions on matrices
Exa 23.2 Calculating mean of of statistical data performing iter-
ations matrices
Exa 23.3 Analysis of statistical data performing iterations on ma-
trices
Exa 23.4 Analysis of statistical data
Exa 23.5 Finding the missing frequency of given statistical data
using given constants
Exa 23.6 Calculating average speed
Exa 23.7 Calculating mean and standard deviation performing it-
erations on matrices
Exa 23.8 Calculating mean and standard deviation performing it-
erations on matrices
10

12. Exa 23.9 Analysis of statistical data performing iterations on ma-
trices
Exa 23.10 Calculating mean and standard deviation of dierent
statistical data when put together
Exa 23.12 Calculating median and quartiles of given statistical data
performing iterations on matrices
Exa 23.13 Calculating coecient of correlation
Exa 24.1 nding the roots of equation
Exa 24.3 nding the roots of equation by the method of false
statement
Exa 24.4 nding rea roots of equation by regula falsi method
Exa 24.5 real roots of equation by newtons method
Exa 24.6 real roots of equation by newtons method
Exa 24.7 evaluating square root by newtons iterative method
Exa 24.10 solving equations by guass elimination method
Exa 24.12 solving equations by guass elimination method
Exa 24.13 solving equations by guass elimination method
Exa 26.2 nding dierence equation
Exa 26.3 solving dierence equation
Exa 26.4 solving dierence equation
Exa 26.6 rming bonacci dierence equation
Exa 26.7 solving dierence equation
Exa 26.8 solving dierence equation
Exa 26.10 solving dierence equation
Exa 26.11 solving dierence equation
Exa 26.12 solving simultanious dierence equation
Exa 26.15.2Z transform
Exa 26.16 evaluating u2 and u3
Exa 27.1 solving ODE with picards method
Exa 27.2 solving ODE with picards method
Exa 27.5 solving ODE using Eulers method
Exa 27.6 solving ODE using Eulers method
Exa 27.7 solving ODE using Modied Eulers method
Exa 27.8 solving ODE using Modied Eulers method
Exa 27.9 solving ODE using Modied Eulers method
Exa 27.10 solving ODE using runge method
Exa 27.11 solving ODE using runge kutta method
Exa 27.12 solving ODE using runge kutta method
11

13. Exa 27.13 solving ODE using runge kutta method
Exa 27.14 solving ODE using milnes method
Exa 27.15 solving ODE using runge kutta and milnes method
Exa 27.16 solving ODE using adamsbashforth method
Exa 27.17 solving ODE using runge kutta and adams method
Exa 27.18 solving simultanious ODE using picards method
Exa 27.19 solving ssecond ODE using runge kutta method
Exa 27.20 solving ODE using milnes method
Exa 28.1 classication of partial dierential equation
Exa 28.2 solving elliptical equation
Exa 28.3 evaluating function satisfying laplace equation
Exa 28.4 solution of poissons equation
Exa 28.5 solving parabolic equation
Exa 28.6 solving heat equation
Exa 28.7 solving wave equation
Exa 28.8 solving wave equation
Exa 34.1 Calculating probability
Exa 34.2.1 Calculating the number of permutations
Exa 34.2.2 Number of permutations
Exa 34.3.1 Calculating the number of committees
Exa 34.3.2 Finding the number of committees
Exa 34.3.3 Finding the number of committees
Exa 34.4.1 Finding the probability of getting a four in a single
throw of a die
Exa 34.4.2 Finding the probability of getting an even number in a
single throw of a die
Exa 34.5 Finding the probability of 53 sundays in a leap year
Exa 34.6 probability of getting a number divisible by 4 under
given conditions
Exa 34.7 Finding the probability
Exa 34.8 Finding the probability
Exa 34.9.1 Finding the probability
Exa 34.9.2 Finding the probability
Exa 34.9.3 Finding the probability
Exa 34.13 probability of drawing an ace or spade from pack of 52
cards
Exa 34.14.1Finding the probability
Exa 34.15.1Finding the probability
12

14. Exa 34.15.2Finding the probability
Exa 34.15.3Finding the probability
Exa 34.16 Finding the probability
Exa 34.17 Finding the probability
Exa 34.18 Finding the probability
Exa 34.19.1Finding the probability
Exa 34.19.2Finding the probability
Exa 34.19.3Finding the probability
Exa 34.20 Finding the probability
Exa 34.22 Finding the probability
Exa 34.23 Finding the probability
Exa 34.25 nding the probability
Exa 34.26 nding the probability
Exa 34.27 nding the probability
Exa 34.28 nding the probability
Exa 34.29 nding the probability
Exa 34.30 nding the probability
Exa 34.31 nding the probability
Exa 34.33 nding the probability
Exa 34.34 nding the probability
Exa 34.35 nding the probability
Exa 34.38 nding the probability
Exa 34.39 nding the probability
Exa 34.40 nding the probability
Exa 35.1 calculating the SD of given sample
Exa 35.2 Calculating SD of sample
Exa 35.3 Analysis of sample
Exa 35.4 Analysis of sample
Exa 35.5 Checking whether real dierence will be hidden
Exa 35.6 Checking whether given sample can be regarded as a
random sample
Exa 35.9 Checking whethet samples can be regarded as taken
from the same population
Exa 35.10 calculating SE of dierence of mean hieghts
Exa 35.12 Mean and standard deviation of a given sample
Exa 35.13 Mean and standard deviation of a given sample
Exa 34.15 Standard deviation of a sample
13

15. List of Figures
1.1 Finding the solution of equation by drawing graphs
1.2 Finding the solution of equation by drawing graphs
1.3 Finding the solution of equation by drawing graphs
6.1 Calculating the area under two curves
14

16. Chapter 1
Solution of equation and curve
tting
Scilab code Exa 1.1nding the roots of quadratic equations
1clear
2clc
3x= ' x ');
4p=2*(x^3)+x^2 -13*x+6
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.2nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=3*(x^3) -4*(x^2)+x+88
5disp " the r o o t s of above equation are ")
6roots
15

17. Scilab code Exa 1.3nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=x^3 -7*(x^2) +36
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.6nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=x^4 -2*(x^3) -21*(x^2) +22*x+40
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.7nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=2*(x^4) -15*(x^3) +35*(x^2) -30*x+8
5disp " the r o o t s of above equation are ")
6roots
16

18. Scilab code Exa 1.11forming an equation with known roots
1clear
2clc
3x= ' x ');
4x1= ' x1 ');
5x2= ' x2 ');
6x3= ' x3 ');
7p=x^3 -3*(x^2)+1
8disp " the r o o t s of above equation are ")
9roots
10disp " l e t ")
11x1 =0.6527036
12x2 = -0.5320889
13x3 =2.8793852
14disp " so the equation whose r o o t s are cube of the
r o o t s of above equation i s ( xx1 ^3)( xx2 ^3)( x
x3 ^3)=0 =>")
15p1=(x-x1^3)*(x-x2^3)*(x-x3^3)
Scilab code Exa 1.12forming an equation under restricted conditions
1clear
2clc
3x= ' x ');
4x1= ' x1 ');
5x2= ' x2 ');
6x3= ' x3 ');
7x4= ' x4 ');
8x5= ' x5 ');
9x6= ' x6 ');
10p=x^3 -6*(x^2) +5*x+8
11disp " the r o o t s of above equation are ")
12roots
13disp " l e t ")
17

19. 14x1 = -0.7784571
15x2 =2.2891685
16x3 =4.4892886
17disp " now , s i n c e we want equation whose sum of
r o o t s i s 0 . sum of r o o t s of above equation i s 6 , so
we w i l l d e c r e a s e ")
18disp " value of each root by 2 i . e . x4=x1 2 ")
19x4=x1 -2
20disp "x5=x22")
21x5=x2 -2
22disp "x6=x32")
23x6=x3 -2
24disp " hence , the r e q u i r e d equation i s ( xx4 )( xx5 )(
xx6 )=0 >")
25p1=(x-x4)*(x-x5)*(x-x6)
Scilab code Exa 1.13nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=6*(x^5) -41*(x^4) +97*(x^3) -97*(x^2) +41*x-6
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.14nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=6*(x^6) -25*(x^5) +31*(x^4) -31*(x^2) +25*x-6
5disp " the r o o t s of above equation are ")
6roots
18

20. Scilab code Exa 1.15nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=x^3 -3*(x^2) +12*x+16
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.16nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=28*(x^3) -9*(x^2)+1
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.17nding the roots of equation containing one variable
1clear
2clc
3x= ' x ');
4p=x^3+x^2 -16*x+20
5disp " the r o o t s of above equation are ")
6roots
19

21. Scilab code Exa 1.18Finding the roots of equation containing one vari-
able
1clear
2clc
3x= ' x ');
4p=x^3 -3*(x^2)+3
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.19Finding the roots of equation containing one vari-
able
1clear
2clc
3x= ' x ');
4p=x^4 -12*(x^3) +41*(x^2) -18*x-72
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.20Finding the roots of equation containing one vari-
able
1clear
2clc
3x= ' x ');
4p=x^4 -2*(x^3) -5*(x^2) +10*x-3
5disp " the r o o t s of above equation are ")
6roots
20

22. Scilab code Exa 1.21Finding the roots of equation containing one vari-
able
1clear
2clc
3x= ' x ');
4p=x^4 -8*(x^2) -24*x+7
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.22Finding the roots of equation containing one vari-
able
1clear
2clc
3x= ' x ');
4p=x^4 -6*(x^3) -3*(x^2) +22*x-6
5disp " the r o o t s of above equation are ")
6roots
Scilab code Exa 1.23Finding the solution of equation by drawing graphs
1clear
2clc
3xset ' window ',1)
4xtitle "My Graph","X a x i s ","Y a x i s ")
5x=
6y1=3-x
7y2=%e^(x-1)
8plot "o")
9plot "+")
10legend("3x","%e^( x1)")
21

23. Figure 1.1: Finding the solution of equation by drawing graphs
11disp " from the graph , i t i s c l e a r that the point of
i n t e r s e c t i o n i s n e a r l y x=1.43 ")
Scilab code Exa 1.24Finding the solution of equation by drawing graphs
1clear
2clc
3xset ' window ',2)
4xtitle "My Graph","X a x i s ","Y a x i s ")
5x=
6y1=x
7y2=
8plot "o")
9plot "+")
10legend("x"," s i n ( x )+%pi/2 ")
11disp " from the graph , i t i s c l e a r that the point of
i n t e r s e c t i o n i s n e a r l y x=2.3 ")
22

24. Figure 1.2: Finding the solution of equation by drawing graphs
Scilab code Exa 1.25Finding the solution of equation by drawing graphs
1clear
2clc
3xset ' window ',3)
4xtitle "My Graph","X a x i s ","Y a x i s ")
5x=
6y1=-sec(x)
7y2=
8plot "o")
9plot "+")
10legend("sec ( x ) "," cosh ( x ) ")
11disp " from the graph , i t i s c l e a r that the point of
i n t e r s e c t i o n i s n e a r l y x=2.3 ")
23

25. Figure 1.3: Finding the solution of equation by drawing graphs
24

26. Chapter 2
Determinants and Matrices
Scilab code Exa 2.1Calculating Determinant
1clc
2syms a;
3syms h;
4syms g;
5syms b;
6syms f;
7syms c;
8A=[a h g;h b f;g f c]
9det
Scilab code Exa 2.2Calculating Determinant
1clear
2clc
3a=[0 1 2 3;1 0 3 0;2 3 0 1;3 0 1 2]
4disp " determinant of a i s ")
5det
25

27. Scilab code Exa 2.3Calculating Determinant
1clc
2syms a;
3syms b;
4syms c;
5A=[a a^2 a^3-1;b b^2 b^3-1;c c^2 c^3-1]
6det
Scilab code Exa 2.4Calculating Determinant
1clear
2clc
3a=[21 17 7 10;24 22 6 10;6 8 2 3;6 7 1 2]
4disp " determinant of a i s ")
5det
Scilab code Exa 5.8Partial derivative of given function
1clc
2syms x y
3u=x^y
4a=
5b=
6c=
7d=
8e=
9f=
10disp ' c l e a r l y , c=f ')
26

28. Scilab code Exa 2.16product of two matrices
1clear
2clc
3A=[0 1 2;1 2 3;2 3 4]
4B=[1 -2;-1 0;2 -1]
5disp "AB= ")
6A*B
7disp "BA= ")
8B*A
Scilab code Exa 2.17Product of two matrices
1clear
2clc
3A=[1 3 0;-1 2 1;0 0 2]
4B=[2 3 4;1 2 3;-1 1 2]
5disp "AB= ")
6A*B
7disp "BA= ")
8B*A
9disp " c l e a r l y AB i s not equal to BA")
Scilab code Exa 2.18Product and inverse of matrices
1clear
2clc
3A=[3 2 2;1 3 1;5 3 4]
4C=[3 4 2;1 6 1;5 6 4]
27

29. 5disp "AB=C >B=inv (A)C")
6B=
Scilab code Exa 2.19Solving equation of matrices
1clear
2clc
3A=[1 3 2;2 0 -1;1 2 3]
4I=
5disp "A^34A^23A+11 I=")
6A^3-4*A*A-3*A+11*I
Scilab code Exa 2.20Nth power of a given matrix
1clc
2A=[11 -25;4 -9]
3n= ' Enter the value of n ") ;
4disp ( 'calculating A^n ' ) ;
5A^n
Scilab code Exa 2.23Inverse of matrix
1clear
2clc
3A=[1 1 3;1 3 -3;-2 -4 -4]
4disp " i n v e r s e of A i s ")
5inv
28

30. Scilab code Exa 2.24.1Rank of a matrix
1clear
2clc
3A=[1 2 3;1 4 2;2 6 5]
4disp "Rank of A i s ")
5rank
Scilab code Exa 2.24.2Rank of a matrix
1clear
2clc
3A=[0 1 -3 -1;1 0 1 1;3 1 0 2;1 1 -2 0]
4disp "Rank of A i s ")
5rank
Scilab code Exa 2.25Inverse of matrix
1clear
2clc
3A=[1 1 3;1 3 -3;-2 -4 -4]
4disp " i n v e r s e of A i s ")
5inv
Scilab code Exa 2.26eigen values vectors rank of matrix
1clear
2clc
3A=[2 3 -1 -1;1 -1 -2 -4;3 1 3 -2;6 3 0 -7]
4[R P]=
29

31. 5disp " rank of A")
6rank
Scilab code Exa 2.28Inverse of a matrix
1clear
2clc
3A=[1 1 1;4 3 -1;3 5 3]
4disp " i n v e r s e of A =")
5inv
Scilab code Exa 2.31Solving equation using matrices
1clear
2clc
3disp " the e q u a t io n s can be re w r i t t e n as AX=B where
X=[x1 ; x2 ; x3 ; x4 ] and ")
4A=[1 -1 1 1;1 1 -1 1;1 1 1 -1;1 1 1 1]
5B=[2; -4;4;0]
6disp " determinant of A=")
7det
8disp " i n v e r s e of A =")
9inv
10disp "X=")
11inv
Scilab code Exa 2.32Solving equation using matrices
1clear
2clc
30

32. 3disp " the e q u a t io n s can be re w r i t t e n as AX=B where
X=[x ; y ; z ] and ")
4A=[5 3 7;3 26 2;7 2 10]
5B=[4;9;5]
6disp " determinant of A=")
7det
8disp " Since det (A) =0, hence , t h i s system of equation
w i l l have i n f i n i t e s o l u t i o n s . . hence , the system i s
c o n s i s t e n t ")
Scilab code Exa 2.34.1predicting nature of equation using rank of matrix
1clc
2A=[1 2 3;3 4 4;7 10 12]
3disp ' rank of A i s ')
4p=
5if
6 disp ' e q u a t i o ns have only a t r i v i a l s o l u t i o n : x=y=z
=0 ')
7else
8 disp ' e q u a t i o ns have i n f i n i t e no . of s o l u t i o n s . ')
9 end
Scilab code Exa 2.34.2predicting nature of equation using rank of matrix
1clc
2A=[4 2 1 3;6 3 4 7;2 1 0 1]
3disp ' rank of A i s ')
4p=
5if
6 disp ' e q u a t i o ns have only a t r i v i a l s o l u t i o n : x=y=z
=0 ')
7else
31

33. 8 disp ' e q u a t i o n s have i n f i n i t e no . of s o l u t i o n s . ')
9 end
Scilab code Exa 2.38Inverse of a matrix
1clear
2clc
3disp " the given e q u a t i o n s can be w r i t t e n as Y=AX
where ")
4A=[2 1 1;1 1 2;1 0 -2]
5disp " determinant of A i s ")
6det
7disp " since , i t s nons i n g u l a r , hence t r a n s f o r m a t i o n i s
r e g u l a r ")
8disp " i n v e r s e of A i s ")
9inv
Scilab code Exa 2.39Transpose and product of matrices
1clear
2clc
3A=[ -2/3 1/3 2/3;2/3 2/3 1/3;1/3 -2/3 2/3]
4disp "A t r a n s p o s e i s equal to ")
5A'
6disp "A( t r a n s p o s e of A)=")
7A*A'
8disp " hence ,A i s orthogonal ")
Scilab code Exa 2.42eigen values and vectors of given matrix
32

34. 1clear
2clc
3A=[5 4;1 2]
4disp " l e t R r e p r e s e n t s the matrix of t r a n s f o r m a t i o n
and P r e p r e s e n t s a d i a g o n a l matrix whose v a l u e s
are the e i g e n v a l u e s of A. then ")
5[R P]=
6disp "R i s normalised . l e t U r e p r e s e n t s unnormalised
v e r s i o n of r ")
7U(:,1)=R(:,1)*
8U(:,2)=R(:,2)*
9disp "two e i g e n v e c t o r s are the two columns of U")
Scilab code Exa 2.43eigen values and vectors of given matrix
1clear
2clc
3A=[1 1 3;1 5 1;3 1 1]
4disp " l e t R r e p r e s e n t s the matrix of t r a n s f o r m a t i o n
and P r e p r e s e n t s a d i a g o n a l matrix whose v a l u e s
are the e i g e n v a l u e s of A. then ")
5[R P]=
6disp "R i s normalised . l e t U r e p r e s e n t s unnormalised
v e r s i o n of r ")
7U(:,1)=R(:,1)*
8U(:,2)=R(:,2)*
9U(:,3)=R(:,3)*
10disp " t h r e e e i g e n v e c t o r s are the t h r e e columns of U
")
Scilab code Exa 2.44eigen values and vectors of given matrix
1clear
33

35. 2clc
3A=[3 1 4;0 2 6;0 0 5]
4disp " l e t R r e p r e s e n t s the matrix of t r a n s f o r m a t i o n
and P r e p r e s e n t s a d i a g o n a l matrix whose v a l u e s
are the e i g e n v a l u e s of A. then ")
5[R P]=
6disp "R i s normalised . l e t U r e p r e s e n t s unnormalised
v e r s i o n of r ")
7U(:,1)=R(:,1)*
8U(:,2)=R(:,2)*
9U(:,3)=R(:,3)*
10disp " t h r e e e i g e n v e c t o r s are the t h r e e columns of U
")
Scilab code Exa 2.45eigen values and characteristic equation
1clear
2clc
3x= ' x ')
4A=[1 4;2 3]
5I=
6disp " e i g e n v a l u e s of A are ")
7spec
8disp " l e t ")
9a=-1;
10b=5;
11disp " hence , the c h a r a c t e r i s t i c equation i s ( xa ) ( xb
) ")
12p=(x-a)*(x-b)
13disp "A^24A5I=")
14A^2-4*A-5*I
15disp " i n v e r s e of A= ")
16inv
34

36. Scilab code Exa 2.46eigen values and characteristic equation
1clear
2clc
3x= ' x ')
4A=[1 1 3;1 3 -3;-2 -4 -4]
5disp " e i g e n v a l u e s of A are ")
6spec
7disp " l e t ")
8a=4.2568381;
9b=0.4032794;
10c= -4.6601175;
11disp " hence , the c h a r a c t e r i s t i c equation i s ( xa ) ( xb
) ( xc ) ")
12p=(x-a)*(x-b)*(x-c)
13disp " i n v e r s e of A= ")
14inv
Scilab code Exa 2.47eigen values and characteristic equation
1clear
2clc
3x= ' x ')
4A=[2 1 1;0 1 0;1 1 2]
5I=
6disp " e i g e n v a l u e s of A are ")
7spec
8disp " l e t ")
9a=1;
10b=1;
11c=3;
35

37. 12disp " hence , the c h a r a c t e r i s t i c equation i s ( xa ) ( xb
) ( xc ) ")
13p=(x-a)*(x-b)*(x-c)
14disp "A^85A^7+7A^63A^5+A^45A^3+8A^22A+I ="
)
15A^8-5*A^7+7*A^6-3*A^5+A^4-5*A^3+8*A^2-2*A+I
Scilab code Exa 2.48eigen values and vectors of given matrix
1clear
2clc
3A=[-1 2 -2;1 2 1;-1 -1 0]
4disp "R i s matrix of t r a n s f o r m a t i o n and D i s a
d i a g o n a l matrix ")
5[R D]=
Scilab code Exa 2.49eigen values and vectors of given matrix
1clear
2clc
3A=[1 1 3;1 5 1;3 1 1]
4disp "R i s matrix of t r a n s f o r m a t i o n and D i s a
d i a g o n a l matrix ")
5[R D]=
6disp "R i s normalised , l e t P denotes unnormalised
v e r s i o n of R. Then ")
7P(:,1)=R(:,1)*
8P(:,2)=R(:,2)*
9P(:,3)=R(:,3)*
10disp "A^4=")
11A^4
36

38. Scilab code Exa 2.50eigen values and vectors of given matrix
1clear
2clc
3disp "3x^2+5y^2+3z^22yz+2zx2xy")
4disp "The matrix of the given q u a d r a t i c form i s ")
5A=[3 -1 1;-1 5 -1;1 -1 3]
6disp " l e t R r e p r e s e n t s the matrix of t r a n s f o r m a t i o n
and P r e p r e s e n t s a d i a g o n a l matrix whose v a l u e s
are the e i g e n v a l u e s of A. then ")
7[R P]=
8disp " so , c a n o n i c a l form i s 2x^2+3y^2+6z ^2")
Scilab code Exa 2.51eigen values and vectors of given matrix
1clear
2clc
3disp "2x1x2+2x1x32x2x3 ")
4disp "The matrix of the given q u a d r a t i c form i s ")
5A=[0 1 1;1 0 -1;1 -1 0]
6disp " l e t R r e p r e s e n t s the matrix of t r a n s f o r m a t i o n
and P r e p r e s e n t s a d i a g o n a l matrix whose v a l u e s
are the e i g e n v a l u e s of A. then ")
7[R P]=
8disp " so , c a n o n i c a l form i s2x^2+y^2+z ^2 ")
Scilab code Exa 2.52Hermitian matrix
1clear
37

39. 2clc
3A=[2+ %i 3 -1+3*%i;-5 %i 4-2*%i]
4disp "A=")
5A'
6disp "AA=")
7A*A'
8disp " c l e a r l y ,AAi s hermitian matrix ")
Scilab code Exa 2.53tranpose and inverse of complex matrix
1clear
2clc
3A=[(1/2) *(1+ %i) (1/2) *(-1+%i);(1/2) *(1+ %i) (1/2) *(1-
%i)]
4disp "A=")
5A'
6disp "AA=")
7A*A'
8disp "AA=")
9A'*A
10disp " i n v e r s e of A i s ")
11inv
Scilab code Exa 2.54Unitary matrix
1clear
2clc
3A=[0 1+2* %i; -1+2*%i 0]
4I=
5disp " IA= ")
6I-A
7disp " i n v e r s e of ( I+A)= ")
8inv
38

40. 9disp " ( ( IA) ( i n v e r s e ( I+A) ) )(( IA) ( i n v e r s e ( I+A) ) )=")
10(((I-A)*(
11disp " ( ( IA) ( i n v e r s e ( I+A) ) ) ( ( IA) ( i n v e r s e ( I+A) ) )=")
12((I-A)*(
13disp " c l e a r l y , the product i s an i d e n t i t y matrix .
hence , i t i s a un ita ry matrix ")
39

41. Chapter 4
Dierentiation and
Applications
Scilab code Exa 4.4.1nding nth derivative
1//
2//
3//
4//
5//
6//
7clc
8disp ' we have to f i n d yn f o r F=cosxcos2xcos3x ' );
9syms x
10F=
11n= ' Enter the order of d i f f e r e n t i a t i o n ") ;
12disp ( 'calculating yn ' ) ;
13yn=d i f f (F , x , n )
14disp ( 'the expression ' ) ;
15disp ( yn ) ;
40

42. Scilab code Exa 4.5nding nth derivative
1//
2//
3//
4//
5//
6//
7clc
8disp ' we have to f i n d yn f o r F=cosxcos2xcos3x ' );
9syms x
10F=x/((x-1) *(2*x+3));
11n= ' Enter the order of d i f f e r e n t i a t i o n : ") ;
12disp ( 'calculating yn ' ) ;
13yn=d i f f (F , x , n )
14disp ( 'the expression ' ) ;
15disp ( yn ) ;
Scilab code Exa 4.6nding nth derivative
1//
2//
3//
4//
5//
6//
7clc
8disp ' we have to f i n d yn f o r F=cosxcos2xcos3x ' );
9syms x a
10F=x/(x^2+a^2);
11n= ' Enter the order of d i f f e r e n t i a t i o n : ") ;
12disp ( 'calculating yn ' ) ;
13yn=d i f f (F , x , n )
14disp ( 'the expression ' ) ;
15disp ( yn ) ;
41

43. Scilab code Exa 4.7nding nth derivative
1//
2//
3//
4//
5//
6//
7clc
8disp ' we have to f i n d yn f o r F=cosxcos2xcos3x ' );
9syms x a
10F=%e^(x)*(2*x+3) ^3;
11//
12disp ' c a l c u l a t i n g yn ');
13yn=
14disp ' the e x p r e s s i o n f o r yn i s ');
15disp
Scilab code Exa 4.8proving the given dierential equation
1//
2//
3//
4//
5//
6//
7clc
8disp ' y=( s i n ^1)x ) s i g n i n v e r s e x ');
9syms x
10y=(
11disp 'we have to prove (1x ^2) y ( n+2)(2n+1)xy ( n+1)n
^2yn ') ;
42

44. 12//
13disp ' c a l c u l a t i n g yn f o r v a r i o u s v a l u e s of n ');
14for
15
16 F=(1-x^2)*
^2+a^2)*
17 disp
18 disp ' the e x p r e s s i o n f o r yn i s ');
19 disp
20 disp ' Which i s equal to 0 ');
21
22end
23disp ' Hence proved ');
Scilab code Exa 4.9proving the given dierential equation
1//
2//
3//
4//
5//
6//
7clc
8disp ' y=e ^( a ( s i n ^1)x ) ) s i g n i n v e r s e x ');
9syms x a
10y=%e^(a*(
11disp 'we have to prove (1x ^2) y ( n+2)(2n+1)xy ( n+1)(
n^2+a ^2) yn ') ;
12//
13disp ' c a l c u l a t i n g yn f o r v a r i o u s v a l u e s of n ');
14for
15
16 //
17 F=(1-x^2)*
^2+a^2)*
43

45. 18 disp
19 disp ' the e x p r e s s i o n f o r yn i s ');
20 disp
21 disp ' Which i s equal to 0 ');
22
23end
24disp ' Hence proved ');
Scilab code Exa 4.10proving the given dierential equation
1clc
2disp ' y ^(1/m)+y^(1/m)=2x ');
3disp ' OR y ^(2/m)2xy ^(1/m)+1 ');
4disp 'OR y=[x+(x^21) ] ^m and y=[x(x^21) ] ^m ');
5
6syms x m
7disp ' For y=[x+(x^21) ] ^m ');
8y=(x+(x^2-1))^m
9disp 'we have to prove ( x^21)y ( n+2)+(2n+1)xy ( n+1)+(
n^2m^2) yn ') ;
10//
11disp ' c a l c u l a t i n g yn f o r v a r i o u s v a l u e s of n ');
12for
13
14 //
15 F=(x^2-1)*
^2-m^2)*
16 disp
17 disp ' the e x p r e s s i o n f o r yn i s ');
18 disp
19 disp ' Which i s equal to 0 ');
20
21end
22disp ' For y=[x(x^21) ] ^m ');
23y=(x-(x^2-1))^m
44

46. 24disp 'we have to prove ( x^21)y ( n+2)+(2n+1)xy ( n+1)+(
n^2m^2) yn ') ;
25//
26disp ' c a l c u l a t i n g yn f o r v a r i o u s v a l u e s of n ');
27for
28
29 //
30 F=(x^2-1)*
^2-m^2)*
31 disp
32 disp ' the e x p r e s s i o n f o r yn i s ');
33 disp
34 disp ' Which i s equal to 0 ');
35
36end
37disp ' Hence proved ');
Scilab code Exa 4.11verify roles theorem
1clc
2disp ' f o r r o l e s theorem F9x ) should be
d i f f e r e n t i a b l e in ( a , b ) and f ( a )=f ( b ) ');
3disp ' Here f ( x )=s i n ( x ) / e ^x ');
4disp ' ');
5syms x
6y=
7
8y1=
9disp
10disp ' putting t h i s to zero we get tan ( x )=1 i e x=pi /4
');
11disp ' value pi /2 l i e s b/w 0 and pi . Hence r o l e s
theorem i s v e r i f i e d ');
45

47. Scilab code Exa 4.16expansion using maclaurins series
1//
2disp ' Maclaurins s e r i e s ');
3disp ' f ( x )=f ( 0 )+xf1 ( 0 )+x ^2/2!f2 ( 0 )+x ^3/3!f3 ( 0 )
+ . . . . . . ');
4syms x a
5//
6 y=
7//
8n= ' e n t e r the number of e x p r e s s i o n in s e r i e s :
');
9a=1;
10t=
11a=0;
12for
13 y1= ' a ',i-1);
14 t=t+x^(i-1)*
15end
16disp
Scilab code Exa 4.17expanding function as fourier series of sine term
1//
2disp ' Maclaurins s e r i e s ');
3disp ' f ( x )=f ( 0 )+xf1 ( 0 )+x ^2/2!f2 ( 0 )+x ^3/3!f3 ( 0 )
+ . . . . . . ');
4syms x a
5
6 y=%e^(
7 n= ' e n t e r the number of e x p r e s s i o n in s e r i s :
');
46

48. 8 a=0;
9t=
10a=0;
11for
12 y1= ' a ',i-1);
13 t=t+x^(i-1)*
14end
15disp
Scilab code Exa 4.18expansion using maclaurins series
1//
2disp ' Maclaurins s e r i e s ');
3disp ' f ( x )=f ( 0 )+xf1 ( 0 )+x ^2/2!f2 ( 0 )+x ^3/3!f3 ( 0 )
+ . . . . . . ');
4syms x a
5
6 y=
7 n= ' e n t e r the number of d i f f e r e n t i a t i o n
i n v o l v e d in maclaurins s e r i e s : ');
8 a=0;
9t=
10a=0;
11for
12 y1= ' a ',i-1);
13 t=t+x^(i-1)*
14end
15disp
Scilab code Exa 4.19expansion using maclaurins series
1//
2disp ' Maclaurins s e r i e s ');
47

49. 3disp ' f ( x )=f ( 0 )+xf1 ( 0 )+x ^2/2!f2 ( 0 )+x ^3/3!f3 ( 0 )
+ . . . . . . ');
4syms x a b
5
6 y=%e^(a*
7 n= ' e n t e r the number of e x p r e s s i o n in s e r i s :
');
8 b=0;
9t=
10
11for
12 y1= ' b ',i-1);
13 t=t+x^(i-1)*
14end
15disp
Scilab code Exa 4.20expansion using taylors series
1//
2disp ' Advantage of s c i l a b i s that we can c a l c u l a t e
log1 . 1 d i r e c t l y without using Taylor s e r i e s ');
3disp ' Use of t a y l o r s e r i e s are given in subsequent
examples ');
4y=
5disp ' l o g ( 1 . 1 )= ');
6disp
Scilab code Exa 4.21taylor series
1//
2disp ' Taylor s e r i e s ');
3disp ' f ( x+h )=f ( x )+hf1 ( x )+h ^2/2!f2 ( x )+h ^3/3!f3 ( x )
+ . . . . . . ');
48

50. 4disp 'To f i n f the t a y l o r expansion of tan1(x+h ) ')
5syms x h
6
7 y=
8 n= ' e n t e r the number of e x p r e s s i o n in s e r i s :
');
9
10t=y;
11
12for
13 y1= ' x ',i-1);
14 t=t+h^(i-1)*(y1)/factorial(i-1);
15end
16disp
Scilab code Exa 4.22evaluating limit
1//
2disp ' Here we need to f i n d f i n d the l i m i t of f ( x ) at
x=0 ')
3syms x
4y=(x*%e^x-
5//
6//
7//
8f=1;
9while
10yn=x*%e^x-
11yd=x^2;
12yn1= ' x ',1);
13yd1= ' x ',1);
14x=0;
15a=
16b=
17if
49

51. 18 yn=yn1;
19 yd=yd1;
20else
21 f=0;
22
23end
24end
25h=a/b;
26disp
Scilab code Exa 4.32tangent to curve
1//
2disp ' Equation of tangent ');
3syms x a y;
4f=(a^(2/3) -x^(2/3))^(3/2);
5s=
6
7Y1=s*(-x)+y;
8X1=-y/s*x;
9g=x-(Y1 -s*(X1 -x));
10disp ' Equation i s g=0 where g i s ');
11disp
Scilab code Exa 4.34nding equation of normal
1//
2disp ' Equation of tangent ');
3syms x a t y
4xo=a*(
5yo=a*(
6s=
7y=yo+s*(x-xo);
50

52. 8disp ' y= ');
9disp
Scilab code Exa 4.35nding angle of intersection of curve
1//
2disp "The two given curves are x^=4y and y^2=4x
which i n t e r s e c t s at ( 0 , 0 ) and ( 4 , 4 ) ') ;
3disp ( ' f o r ( 4 , 4 ) ') ;
4x=4;
5syms x
6y1=x ^2/4;
7y2=2x ^(1/2) ;
8m1=d i f f ( y1 , x , 1 ) ;
9m2=d i f f ( y2 , x , 1 ) ;
10x=4;
11m1=e v a l (m1) ;
12m2=e v a l (m2) ;
13
14disp ( ' Angle between them i s ( r a d i a n s ) :') ;
15t=atan ( (m1m2) /(1+m1m2) ) ;
16disp ( t ) ;
Scilab code Exa 4.37prove given tangent statement
1//
2syms a t
3x=a*(
4y=a*
5s=
6disp ' length of tangent ');
7l=y*(1+s)^(0.5);
8disp
51

53. 9disp ' checking f o r i t s dependency on t ')
10
11f=1
12t=0;
13k=
14for
15 t=i;
16 if
17 f=0;
18 end
19end
20if
21 disp " v e r i f i e d and equal to a");
22 disp ' subtangent ');
23 m=y/s;
24 disp
Scilab code Exa 4.39nding angle of intersection of curve
1//
2clc
3disp ' Angle of i n t e r s e c t i o n ');
4disp ' point of i n t e r s e c t i o n of r=s i n t+c o s t and r=2
s i n t i s t=pi /4 ');
5disp ' tanu=dQ/ drr ');
6syms Q ;
7
8r1=2*
9r2=
10u=
11Q=%pi /4;
12u=
13disp ' The angle at point of i n t e r s e c t i o n in r a d i a n s
i s : ');
14disp
52

54. Scilab code Exa 4.41nding pedal equation of parabola
1//
2clc
3disp ' tanu=dQ/ drr ');
4syms Q a;
5
6r=2*a/(1-
7
8u=
9u=
10p=r*
11syms r;
12Q=
13
14// 2a
15p=
16disp
Scilab code Exa 4.43nding radius of curvature of cycloid
1//
2syms a t
3x=a*(t+
4y=a*(1-
5s2=
6s1=
7
8r=(1+ s1^2) ^(3/2)/s2;
9disp ' The r a d i u s of curvature i s : ');
10disp
53

55. Scilab code Exa 4.46radius of curvature of cardoid
1//
2disp ' r a d i u s of curvature ');
3syms a t
4r=a*(1-
5r1=
6l=(r^2+r1^2) ^(3/2) /(r^2+2* r1^2-r*r1);
7syms r;
8t=
9l=
10disp
11disp ' Which i s p r o p o r t i o n a l to r ^0.5 ');
Scilab code Exa 4.47cordinates of centre of curvature
1//
2disp ' The c e n t r e of curvature ');
3syms x a y
4y=2*(a*x)^0.5;
5y1=
6y2=
7xx=x-y1 *(1+ y1)^2/y2;
8yy=y+(1+ y1^2)/y2;
9disp ' the c o o r d i n a t e s x , y are resp : ');
10
11disp
12disp
54

56. Scilab code Exa 4.48proof statement cycloid
1//
2disp ' c e n t r e of curvature of given c y c l o i d ');
3syms a t
4x=a*(t-
5y=a*(1-
6y1=
7y2=
8xx=x-y1 *(1+ y1)^2/y2;
9yy=y+(1+ y1^2)/y2;
10
11disp ' the c o o r d i n a t e s x , y are resp : ');
12disp
13disp
14disp ' which another parametric equation of c y c l o i d '
);
Scilab code Exa 4.52maxima and minima
1//
2//
3disp 'To f i n d the maxima and minima of given
f u n c t i o n put f1 ( x )=0 ');
4syms x
5//
6f=3*x^4-2*x^3-6*x^2+6*x+1;
7k=
8x= ' x ');
9k=
Scilab code Exa 4.61nding the asymptotes of curve
55

57. 1//
2clc
3disp ' to f i n d the assymptote of given curve ' );
4syms x y
5f=x^2*y^2-x^2*y-x*y^2+x+y+1;
6//
7f1=coeffs(f,x,2);
8disp ' assymptotes p a r a l l e l to xx i s i s given by f1=0
where f1 i s : ');
9disp
10f2=coeffs(f,y,2);
11disp ' assymptotes p a r a l l e l to ya x i s i s given by f2
=0 and f2 i s : ');
12disp
56

58. Chapter 5
Partial Dierentiation And Its
Applications
Scilab code Exa 5.5Partial derivative of given function
1clc
2syms x y z
3v=(x^2+y^2+z^2) ^( -1/2)
4a=
5b=
6c=
7a+b+c
Scilab code Exa 5.14Partial derivative of given function
1clc
2syms x y
3u=
4a=
5b=
6c=
57

59. 7d=
8e=
9x*a+y*b
10(1/2)*
11(x^2)*c+2*x*y*e+(y^2)*d
12(-
Scilab code Exa 5.25.1Partial derivative of given function
1clc
2syms r l
3x=r*
4y=r*
5a=
6b=
7c=
8d=
9A=[a b;c d]
10det
Scilab code Exa 5.25.2Partial derivative of given function
1clc
2syms r l z
3x=r*
4y=r*
5m=z
6a=
7b=
8c=
9d=
10e=
11f=
58

60. 12g=
13h=
14i=
15A=[a b c;d e f;g h i]
16det
Scilab code Exa 5.25.3Partial derivative of given function
1clc
2syms r l m
3x=r*
4y=r*
5z=r*
6a=
7b=
8c=
9d=
10e=
11f=
12g=
13h=
14i=
15A=[a b c;d e f;g h i]
16det
Scilab code Exa 5.26Partial derivative of given function
1clc
2syms x1 x2 x3
3y1=(x2*x3)/x1
4y2=(x3*x1)/x2
5y3=(x1*x2)/x3
6a=
59

61. 7b=
8c=
9d=
10e=
11f=
12g=
13h=
14i=
15A=[a b c;d e f;g h i]
16det
Scilab code Exa 5.30Partial derivative of given function
1clc
2syms x y
3u=x*(1-y^2) ^0.5+y*(1-x^2) ^0.5
4v=
5a=
6b=
7c=
8d=
9A=[a b; c d ]
10det
60

62. Chapter 6
Integration and its Applications
Scilab code Exa 6.1.1indenite integral
1//
2disp ' I n d e f i n i t e i n t e g r a l ');
3syms x
4f=integ ((
5disp
Scilab code Exa 6.1.2indenite integral
1//
2disp ' I n d e f i n i t e i n t e g r a l ');
3syms x
4f=integ ((
5disp
Scilab code Exa 6.2.1denite integral
61

63. 1//
2disp ' d e f i n i t e i n t e g r a l ');
3syms x
4f=integ ((
5disp
Scilab code Exa 6.2.2Denite Integration of a function
1//
2//
3clc
4disp ' d e f i n i t e i n t e g r a l ');
5syms x a
6g=x^7/(a^2-x^2) ^1/2
7f=integ(g,x,0,a);
8disp
Scilab code Exa 4.2.3denite integral
1//
2//
3clc
4disp ' d e f i n i t e i n t e g r a l ');
5syms x a
6g=x^3*(2*a*x-x^2) ^(1/2);
7f=integ(g,x,0,2*a);
8disp
Scilab code Exa 6.2.3denite integral
62

64. 1//
2//
3clc
4disp ' d e f i n i t e i n t e g r a l ');
5syms x a n
6g=1/(a^2+x^2)^n;
7f=integ(g,x,0,%inf);
8disp
Scilab code Exa 6.4.1denite integral
1//
2clc
3disp ' d e f i n i t e i n t e g r a l ');
4syms x
5g=(
6f=integ(g,x,0,%pi /6);
7disp
Scilab code Exa 4.4.2denite integral
1//
2clc
3disp ' d e f i n i t e i n t e g r a l ');
4syms x
5g=x^4*(1 -x^2) ^(3/2);
6f=integ(g,x,0,1);
7disp
Scilab code Exa 6.5denite integral
63

65. 1//
2//
3clc
4disp ' d e f i n i t e i n t e g r a l ');
5syms x m n
6n= ' Enter n : ');
7m= ' Enter m : ');
8g=(
9f=integ(g,x,0,%pi /2);
10disp
11g2=(
12f2=m/(m+n)*integ(g2 ,x,0,%pi /2);
13disp
14disp ' Equal ');
Scilab code Exa 6.6.1reducing indenite integral to simpler form
1//
2clc
3disp ' d e f i n i t e i n t e g r a l ');
4syms x a
5n= ' Enter n : ');
6g=
7
8f=integ(g,x);
9disp
Scilab code Exa 6.7.1Indenite Integration of a function
1clc
2syms x
3disp
64

66. Scilab code Exa 6.8Getting the manual input of a variable and integra-
tion
1clc
2n= ' Enter the value of n ") ;
3p=i n t e g r a t e ( '( ' x ',0,%pi /4)
4q= ' ( tan ( x ) ) ^( n+1) ',' x ',0,%pi /4)
5disp ' n ( p+q )= ')
6disp
Scilab code Exa 6.9.1Denite Integration of a function
1clear
2clc
3integrate ' sec ( x ) ^4 ',' x ',0,%pi /4)
Scilab code Exa 6.9.2Denite Integration of a function
1clear
2clc
3integrate ' 1/ s i n ( x ) ^3 ',' x ',%pi/3,%pi /2)
Scilab code Exa 6.10denite integral
1
2//
3clc
65

67. 4syms x
5g=x*
6f=integ(g,x,0,%pi);
7disp
Scilab code Exa 6.12Denite Integration of a function
1clear
2clc
3integrate ' s i n ( x ) ^ 0 . 5 / ( s i n ( x ) ^0.5+ cos ( x ) ^ 0 . 5 ) ',' x '
,0,%pi /2)
Scilab code Exa 6.13sum of innite series
1
2//
3clc
4syms x
5disp ' The summation i s e q u i v a l e n t to i n t e g r a t i o n of
1/(1+x ^2) from 0 to 1 ' );
6g=1/(1+x^2);
7f=integ(g,x,0,1);
8disp
Scilab code Exa 6.14nding the limit of the function
1//
2clc
3syms x
66

68. 4disp ' The summation i s e q u i v a l e n t to i n t e g r a t i o n of
l o g (1+x ) from 0 to 1 ');
5g=
6f=integ(g,x,0,1);
7disp
Scilab code Exa 6.15Denite Integration of a function
1clear
2clc
3integrate ' xs i n ( x ) ^8cos ( x ) ^4 ',' x ',0,%pi)
Scilab code Exa 6.16Denite Integration of a function
1clear
2clc
3integrate ' l o g ( s i n ( x ) ) ',' x ',0,%pi /2)
Scilab code Exa 6.24Calculating the area under two curves
1clear
2clc
3xset ' window ',1)
4xtitle "My Graph","X a x i s ","Y a x i s ")
5x=
6y1=(x+8)/2
7y2=x^2/8
8plot "o")
9plot "+")
10legend(" ( x+8)/2 ","x ^2/8 ")
67

69. Figure 6.1: Calculating the area under two curves
11disp " from the graph , i t i s c l e a r that the p o i n t s of
i n t e r s e c t i o n are x=4 and x=8.")
12disp "So , our r e g i o n of i n t e g r a t i o n i s from x=4 to x
=8")
13integrate ' ( x+8)/2x ^2/8 ',' x ',-4,8)
68

70. Chapter 9
Innite Series
Scilab code Exa 9.1to nd the limit at innity
1clc
2syms n;
3f=((1/n)^2 -2*(1/n))/(3*(1/n)^2+(1/n))
4disp
Scilab code Exa 9.1.3to nd the limit at innity
1clc
2syms n;
3f=3+( -1)^n
4limit(f,n,%inf)
Scilab code Exa 9.2.1to nd the sum of series upto innity
1clc
2syms n
69

71. 3disp ' 1+2+3+4+5+6+7+....+n + . . . . . = ')
4p=1/n*(1/n+1)/2
5disp
Scilab code Exa 9.2.2to check for the type of series
1clc
2disp ' 541+541+541+541+.........=0,5,1
according to the no . of terms . ' )
3disp ' c l e a r l y , in t h i s case sum doesnt tend to a
unique l i m i t . hence , s e r i e s i s o s c i l l a t o r y . ')
Scilab code Exa 9.5.1to check the type of innite series
1clc
2syms n;
3v=1/((1/n)^2)
4u=(2/n-1) /(1/n*(1/n+1) *(1/n+2))
5disp
6disp ' both u and v converge and d i v e r g e together ,
hence u i s convergent ')
Scilab code Exa 9.5.2to check the type of innite series
1clc
2syms n;
3v=n
4u=((1/n)^2) /((3/n+1) *(3/n+4) *(3/n+7))
5disp
6disp ' both u and v converge and d i v e r g e together ,
hence u i s d i v e r g e n t ')
70

72. Scilab code Exa 9.7.1to check the type of innite series
1clc
2syms n
3disp ' u=((n+1) ^0.51) / ( ( n+2)^31)=>')
4//
5u=((1+1/(1/n)) -(1/n)^( -0.5))/(((1/n)^5/2) *((1+2/(1/n
))^3 -(1/n)^(-3)))
6v=(1/n)^( -5/2)
7disp
8//
9disp ' s i n c e , v i s convergent , so u i s a l s o
conzavergent . ')
Scilab code Exa 9.7.3to check the type of innite series
1clc
2syms n
3disp
Scilab code Exa 9.8.1to nd the sum of series upto innity
1clc
2syms x n;
3//
4u=(x^(2*(1/n) -2))/(((1/n)+1) *(1/n)^0.5)
5v=(x^(2*(1/n)))/((1/n+2) *(1/n+1) ^0.5)
6disp
71

73. Scilab code Exa 9.8.2to nd the limit at innity
1clc
2syms x n;
3//
4u=((2^(1/n) -2)*(x^(1/n-1)))/(2^(1/n)+1)
5v=((2^((1/n)+1) -2)*(x^(1/n)))/(2^(1/n+1) +1)
6disp
Scilab code Exa 9.10.1to nd the limit at innity
1clc
2syms x n;
3u=1/(1+x^(-n));
4v=1/(1+x^(-n-1));
5disp
Scilab code Exa 9.10.2to nd the limit at innity
1clc
2syms a b n;
3l=(b+1/n)/(a+1/n)
4disp
Scilab code Exa 9.11.1to nd the limit at innity
72

74. 1clc
2syms x n;
3disp ' u = ( ( 4 . 7 . . . . ( 3 n+1) )x^n ) / ( 1 . 2 . . . . . n ) ')
4disp ' v = ( ( 4 . 7 . . . . ( 3 n+4)x ^( n+1) ) / ( 1 . 2 . . . . . ( n+1) ) ')
5disp ' l=u/v=>')
6l=(1+n)/((3+4*n)*x)
7disp
Scilab code Exa 9.11.2to nd the limit at innity
1clc
2syms x n;
3u=((( factorial(n))^2)*x^(2*n))/factorial (2*n)
4v=((( factorial(n+1))^2)*x^(2*(n+1)))/factorial (2*(n
+1))
5limit(u/v,n,%inf)
73

75. Chapter 10
Fourier Series
Scilab code Exa 10.1nding fourier series of given function
1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x
5ao=1/ %pi*integ(
6s=ao/2;
7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9 ai=1/ %pi*integ(
10 bi=1/ %pi*integ(
11 s=s+float(ai)*
12end
13disp
Scilab code Exa 10.2nding fourier series of given function
74

76. 1//
2//
3disp 'To f i n d the f o u r i e r transform of given
f u n c t i o n ');
4syms x s
5F=integ(
6disp
7// >
8F1=integ(
Scilab code Exa 10.3nding fourier series of given function
1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x
5ao=1/ %pi*( integ (-1*%pi*x^0,x,-%pi ,0)+integ(x,x,0,%pi
));
6s=ao/2;
7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9 ai=1/ %pi*( integ (-1*%pi*
x*
10 bi=1/ %pi*( integ (-1*%pi*x^0*
integ(x*
11 s=s+float(ai)*
12end
13disp
Scilab code Exa 10.4nding fourier series of given function
75

77. 1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x l
5ao=1/l*integ(
6s=ao/2
7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9 ai=1/l*integ(
10 bi=1/l*integ(
11 s=s+float(ai)*
/l);
12end
13disp
Scilab code Exa 10.5nding fourier series of given function in interval
minus pi to pi
1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x l
5s=0;
6n= ' e n t e r the no of terms upto each of s i n
terms in the expansion : ' );
7for
8
9 bi=2/ %pi*integ(x*
10 s=s+float(bi)*
11end
12disp
76

78. Scilab code Exa 10.6nding fourier series of given function in interval
minus l to l
1//
2//
3clc
4disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
5syms x l
6ao=2/l*integ(x^2,x,0,l);
7s=float(ao)/2;
8n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
9for
10 ai=2/l*integ(x^2*
11 // i n t e g xs i n x l
12 s=s+float(ai)*
13 end
14disp
Scilab code Exa 10.7nding fourier series of given function in interval
minus pi to pi
1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x
5ao=2/ %pi*( integ(
%pi/2,%pi));
6s=ao/2;
77

79. 7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9 ai=2/ %pi*( integ(
cos
10 // ( 1%pixs i n x 1%pi
i n t e g s i n x
11 s=s+float(ai)*
12end
13disp
Scilab code Exa 10.8nding fourier series of given function in interval
minus pi to pi
1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x
5ao=2/ %pi*( integ ((1 -2*x/%pi),x,0,%pi));
6s=ao/2;
7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9 ai=2/ %pi*( integ ((1 -2*x/%pi)*
10 // ( 1%pixs i n x 1%pi
i n t e g s i n x
11 s=s+float(ai)*
12end
13disp
Scilab code Exa 10.9nding half range sine series of given function
78

80. 1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x l
5
6s=0;
7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9// i n t e g xcos %pix l
10 bi=integ(x*
11 s=s+float(bi)*
12end
13disp
Scilab code Exa 10.10nding half range cosine series of given function
1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x
5ao =2/2*( integ(x,x,0,2));
6s=ao/2;
7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9 ai =2/2*( integ(x*
10 // ( 1%pixs i n x 1%pi
i n t e g s i n x
11 s=s+float(ai)*
12end
13disp
79

81. Scilab code Exa 10.11expanding function as fourier series of sine term
1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x
5ao=0;
6s=ao;
7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9 bi =2/1*( integ ((1/4 -x)*
integ ((x -3/4)*
10 s=s+float(bi)*
11end
12disp
Scilab code Exa 10.12nding fourier series of given function
1//
2clc
3disp ' f i n d i n g the f o u r i e r s e r i e s of given f u n c t i o n ')
;
4syms x
5ao=1/ %pi*integ(x^2,x,-%pi ,%pi);
6s=ao/2;
7n= ' e n t e r the no of terms upto each of s i n or
cos terms in the expansion : ' );
8for
9 ai=1/ %pi*integ ((x^2)*
10 bi=1/ %pi*integ ((x^2)*
80

82. 11 s=s+float(ai)*
12end
13disp
Scilab code Exa 10.13nding complex form of fourier series
1//
2clc
3disp ' The complex form of s e r i e s i s summation of f (n
, x ) where n v a r i e s from%inf to %inf and f (n , x )
i s given by : ');
4syms n x
5cn =1/2* integ(
6fnx=float(cn)*
7
8disp
Scilab code Exa 10.14practical harmonic analysis
1//
2//
1.52
3// %pi %pi %pi
/6 %pi %pi %pi %pi
4disp ' P r a c t i c a l harmonic a n a l y s i s ');
5syms x
6xo= ' Input xo matrix : ');
7yo= ' Input yo matrix : ');
8ao=2*
9s=ao/2;
10n= 'No of s i n or cos term in expansion : ' );
11for
12 an=2*
81

83. 13 bn=2*
14 s=s+float(an)*
15
16 end
17 disp
Scilab code Exa 10.15practical harmonic analysis
1//
2//
3// 0.88.25
4//
5disp ' P r a c t i c a l harmonic a n a l y s i s ');
6syms x T
7xo= ' Input xo matrix ( in f a c t o r of T) : ');
8yo= ' Input yo matrix : ');
9ao=2*
10s=ao/2;
11n= 'No of s i n or cos term in expansion : ' );
12i=1
13 an =2*( yo.*
14 bn =2*( yo.*
15 s=s+float(an)*
*2* %pi/T);
16
17 disp
18 disp ' D i r e c t c u r r e n t : ');
19i=
Scilab code Exa 10.16practical harmonic analysis
1//
2//
82

84. 3// 0.88.25
4//
5disp ' P r a c t i c a l harmonic a n a l y s i s ');
6syms x T
7xo= ' Input xo matrix ( in f a c t o r of T) : ');
8yo= ' Input yo matrix : ');
9ao=2*
10s=ao/2;
11n= 'No of s i n or cos term in expansion : ' );
12i=1
13 an =2*( yo.*
14 bn =2*( yo.*
15 s=s+float(an)*
*2* %pi/T);
16
17 disp
18 disp ' D i r e c t c u r r e n t : ');
19i=
Scilab code Exa 10.17practical harmonic analysis
1//
2//
3// 0.88.25
4//
5disp ' P r a c t i c a l harmonic a n a l y s i s ');
6syms x T
7xo= ' Input xo matrix ( in f a c t o r of T) : ');
8yo= ' Input yo matrix : ');
9ao=2*
10s=ao/2;
11n= 'No of s i n or cos term in expansion : ' );
12i=1
13 an =2*( yo.*
14 bn =2*( yo.*
83

85. 15 s=s+float(an)*
*2* %pi/T);
16
17 disp
18 disp ' D i r e c t c u r r e n t : ');
19i=
84

86. Chapter 13
Linear Dierential Equations
Scilab code Exa 13.1solvinf linear dierential equation
1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4syms c1 c2 x
5m= 'm');
6f=m^2+m-2;
7r=
8disp
9y=0;
10//
11 //
12 // exp x
13 //
14 y=c1*
15 disp ' y= ');
16 disp
Scilab code Exa 13.2solving linear dierential equation
85

87. 1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4syms c1 c2 x;
5m= 'm');
6f=m^2+6*m+9;
7r=
8disp
9disp ' r o o t s are equal so s o l u t i o n i s given by : ');
10disp ' y= ');
11y=(c1+x*c2)*
12disp
Scilab code Exa 13.3solving linear dierential equation
1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4syms c1 c2 c3 x
5m= 'm');
6f=m^3+m^2+4*m+4;
7r=
8disp
9y=c1*
10disp ' y= ');
11disp
Scilab code Exa 13.4solving linear dierential equation
1//
2clc
86

88. 3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4m= 'm');
5syms c1 c2 c3 c4 x
6f=m^4+4;
7r=
8disp
9y=c1*
exp
10disp ' y= ');
11disp
Scilab code Exa 13.5nding particular integral
1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4m= 'm');
5f=m^2+5*m+6;
6//
7y=
8disp 'y');
9disp
Scilab code Exa 13.6nding particular integral
1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4m= 'm');
5f=(m+2)*(m-1) ^2;
87

89. 6r=
7disp
8disp ' y=1/ f (D)[ exp(2x )+exp ( x )exp(x ) ');
9disp ' using 1/ f (D) exp ( ax )=x/ f1 (D)exp ( ax ) i f f (m)=0 '
);
10y1=x*
11y2=
12y3=x^2*
13y=y1+y2+y3;
14disp ' y= ');
15disp
Scilab code Exa 13.7nding particular integral
1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4m= 'm');
5f=m^3+1;
6disp ' Using the i d e n t i t y 1/ f (D^2)s i n ( ax+b ) [ or cos (
ax+b ) ]=1/ f (a ^2)s i n ( ax+b ) [ or cos ( ax+b ) ] t h i s
equation can be reduced to ' );
7disp ' y=(4D+1) /65cos (2 x1) ');
8y=(
9disp ' y= ');
10disp
Scilab code Exa 13.8nding particular integral
1//
2clc
88

90. 3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4m= 'm');
5f=m^3+4*m;
6disp ' using 1/ f (D) exp ( ax )=x/ f1 (D)exp ( ax ) i f f (m)=0 '
);
7disp ' y=x1/(3D^2+4)sin2x ');
8disp ' Using the i d e n t i t y 1/ f (D^2)s i n ( ax+b ) [ or cos (
ax+b ) ]=1/ f (a ^2)s i n ( ax+b ) [ or cos ( ax+b ) ] t h i s
equation can be reduced to ' );
9disp ' y=x/8sin2x ');
10disp ' y= ');
11y=-x*
12disp
Scilab code Exa 13.9nding particular integral
1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4m= 'm');
5
6disp ' y=1/(D(D+1) ) [ x^2+2x+4] can be w r i t t e n as (1D+
D^2) /D[ x^2+2x+4] which i s combination of
d i f f e r e n t i a t i o n and i n t e g r a t i o n ');
7g=x^2+2*x+4;
8f=g-
9y=integ(f,x);
10disp ' y= ');
11disp
Scilab code Exa 13.10nding particular integral
89

91. 1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
Scilab code Exa 13.11solving the given linear equation
1//
2clc
3disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
4disp 'CF + PI ');
5syms c1 c2 x
6m= 'm');
7f=(m-2) ^2;
8r=
9disp
10disp 'CF i s given by ');
11cf=(c1+c2*x)*
12disp
13disp ' ');
14disp ' PI =8f1/(D2) ^ 2 [ exp (2 x ) ]+f1/(D2) ^ 2 [ s i n (2 x )
]+f1/(D2) ^ 2 [ x ^2]g');
15disp ' using i d e n t i t i e s i t reduces to : ');
16pi=4*x^2*
17disp
18y=cf+pi;
19disp ' The s o l u t i o n i s : y= ');
20disp
Scilab code Exa 13.12solving the given linear equation
1//
90

92. 2clc
3
4disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
5disp 'CF + PI ');
6syms c1 c2 x
7m= 'm');
8f=(m^2-4);
9r=
10disp
11disp 'CF i s given by ');
12cf=c1*
13disp
14disp ' ');
15disp ' PI =8f1/(D^24) [ xsinh ( x ) ] ');
16disp ' using i d e n t i t i e s i t reduces to : ');
17pi=-x/6*(
18disp
19y=cf+pi;
20disp ' The s o l u t i o n i s : y= ');
21disp
Scilab code Exa 13.13solving the given linear equation
1//
2clc
3
4disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
5disp 'CF + PI ');
6syms c1 c2 x
7m= 'm');
8f=(m^2-1);
9r=
10disp
91

93. 11disp 'CF i s given by ');
12cf=c1*
13disp
14disp ' ');
15disp ' PI =f1/(D^21) [ xs i n (3 x )+cos ( x ) ] ');
16disp ' using i d e n t i t i e s i t reduces to : ');
17pi = -1/10*(x*
18disp
19y=cf+pi;
20disp ' The s o l u t i o n i s : y= ');
21disp
Scilab code Exa 13.14solving the given linear equation
1//
2clc
3
4disp ' s o l u t i o n of the given l i n e a r d i f f e r e n t i a l
equation i s given by : ' );
5disp 'CF + PI ');
6syms c1 c2 c3 c4 x
7m= 'm');
8f=(m^4+2*m^2+1);
9r=
10disp
11disp 'CF i s given by ');
12cf=
;
13disp
14disp ' ');
15disp ' PI =f1/(D^4+2D+1) [ x^2cos ( x ) ] ');
16disp ' using i d e n t i t i e s i t reduces to : ');
17pi = -1/48*((x^4-9*x^2)*
18disp
19y=cf+pi;
92

94. 20disp ' The s o l u t i o n i s : y= ');
21disp
93

95. Chapter 21
Laplace Transform
Scilab code Exa 21.1.1nding laplace transform
1//
2disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
3syms t s
4disp
Scilab code Exa 21.1.2nding laplace transform
1//
2disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
3syms t s
4disp
Scilab code Exa 21.1.3nding laplace transform
1//
2disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
94

96. 3syms t s
4disp
Scilab code Exa 21.2.1nding laplace transform
1//
2disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
3syms t s
4f=
5disp
Scilab code Exa 21.2.2nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s
5f=
6disp
Scilab code Exa 21.2.3nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s
5f=
6disp
95

97. Scilab code Exa 21.4.1nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s a
5f=t*
6disp
Scilab code Exa 21.4.2nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s a
5f=t*
6l=laplace(f,t,s);
7disp
Scilab code Exa 21.5nding laplace transform
1//
2//
3clc
4syms t s u
5f=integ(
%inf);
6disp
96

98. Scilab code Exa 21.7nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s a
5f=
6disp
Scilab code Exa 21.8.1nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s a
5f=t*
6disp
Scilab code Exa 21.8.2nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s a
5f=t^2*
6disp
97

99. Scilab code Exa 21.8.3nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s a
5f=
6l=laplace(f,t,s)
7disp
Scilab code Exa 21.8.4nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s a
5f=
6l=laplace(f,t,s)
7disp
Scilab code Exa 21.9.1nding laplace transform
1//
2//
3clc
4disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
5syms t s a
6f=(1-
7
8l=laplace(f,t,s)
9disp
98

100. Scilab code Exa 21.9.2nding laplace transform
1//
2clc
3disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
4syms t s a b
5f=(
6
7l=laplace(f,t,s)
8disp
Scilab code Exa 21.10.1nding laplace transform
1//
2clc
3disp 'To f i n d the the given i n t e g r a l f i n d the
l a p l a c e of t s i n ( t ) and put s=2 ');
4syms t s m
5f=
6
7l=laplace(f,t,s)
8s=2
9
10disp
Scilab code Exa 21.10.3nding laplace transform
1//
2//
99

101. 3clc
4disp 'To f i n d the l a p l a c e of given f u n c t i o n in t ');
5syms t s a b
6f=integ(
7
8l=laplace(f,t,s)
9disp
Scilab code Exa 21.11.1nding inverse laplace transform
1//
2disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
3syms s t
4f=(s^2-3*s+4)/s^3;
5il=ilaplace(f,s,t);
6disp
Scilab code Exa 21.11.2nding inverse laplace transform
1//
2disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
3syms s t
4f=(s+2) /(2*s^2-4*s+13));
5il=ilaplace(f,s,t);
6disp
Scilab code Exa 21.12.1nding inverse laplace transform
100

102. 1//
2disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
3syms s t
4f=((2*s^2-6*s+5)/(s^3-6*s^2+11*s-6);
5il=ilaplace(f,s,t);
6disp
Scilab code Exa 21.12.3nding inverse laplace transform
1//
2disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
3syms s t
4f=(4*s+5) /((s-1) ^2*(s+2));
5il=ilaplace(f,s,t);
6disp
Scilab code Exa 21.13.1nding inverse laplace transform
1//
2disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
3syms s t
4f=(5*s+3) /((s-1)*(s^2+2*s+5));
5il=ilaplace(f,s,t);
6disp
Scilab code Exa 21.13.2nding inverse laplace transform
101

103. 1//
2//
3
4disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
5syms s t a
6f=s/(s^4+4*a^4);
7il=ilaplace(f,s,t);
8disp
Scilab code Exa 21.14.1nding inverse laplace transform
1
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5f=s^2/(s-2) ^3;
6il=ilaplace(f,s,t);
7disp
Scilab code Exa 21.14.2nding inverse laplace transform
1
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5f=(s+3) /((s^2-4*s+13));
6il=ilaplace(f,s,t);
7disp
102

104. Scilab code Exa 21.15.1nding inverse laplace transform
1//
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5f=1/(s*(s^2+a^2));
6il=ilaplace(f,s,t);
7disp
Scilab code Exa 21.15.2nding inverse laplace transform
1
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5f=1/(s*(s+a)^3);
6il=ilaplace(f,s,t);
7disp
Scilab code Exa 21.16.1nding inverse laplace transform
1//
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
103

105. 5f=s/((s^2+a^2) ^2);
6il=ilaplace(f,s,t);
7disp
Scilab code Exa 21.16.2nding inverse laplace transform
1//
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5f=s^2/((s^2+a^2) ^2);
6il=ilaplace(f,s,t);
7disp
Scilab code Exa 21.16.3nding inverse laplace transform
1//
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5
6f=1/((s^2+a^2) ^2);
7il=ilaplace(f,s,t);
8disp
Scilab code Exa 21.17.1nding inverse laplace transform
1//
104

106. 2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5
6f=(s+2)/(s^2*(s+1)*(s-2));
7il=ilaplace(f,s,t);
8disp
Scilab code Exa 21.17.2nding inverse laplace transform
1//
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5
6f=(s+2)/(s^2+4*s+5) ^2;
7il=ilaplace(f,s,t);
8disp
Scilab code Exa 21.19.1nding inverse laplace transform
1//
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a
5
6f=s/(s^2+a^2) ^2;
7il=ilaplace(f,s,t);
8disp
105

107. Scilab code Exa 21.19.2nding inverse laplace transform
1//
2//
3disp 'To f i n d the i n v e r s e l a p l a c e transform of the
f u n c t i o n ');
4syms s t a b
5
6f=s^2/((s^2+a^2)*(s^2+b^2));
7il=ilaplace(f,s,t);
8disp
Scilab code Exa 21.28.1nding laplace transform
1//
2syms s t
3f=integ(
,t,2,3);
4disp ' Laplace of given f u n c t i o n i s ');
5disp
Scilab code Exa 21.28.2nding laplace transform
1//
2syms s t
3f=integ(
4disp ' Laplace of given f u n c t i o n i s ');
5disp
106

108. Scilab code Exa 21.34nding laplace transform
1//
2//
3disp ' to f i n d the l a p l a c e transform of p e r i o d i c
f u n c t i o n ');
4syms w t s
5f=1/(1 -
,0,%pi/w);
6disp
107

109. Chapter 22
Integral Transform
Scilab code Exa 22.1nding fourier sine integral
1//
2//
3disp 'To f i n d the f o u r i e r s i n e i n t e g r a l ');
4syms x t u
5fs=2/ %pi*integ(
),t,0,%inf));
6disp
Scilab code Exa 22.2nding fourier transform
1//
2//
3disp 'To f i n d the f o u r i e r transform of given
f u n c t i o n ');
4syms x s
5F=integ(
6disp
7// >
8F1=integ(
108

110. Scilab code Exa 22.3nding fourier transform
1//
2//
3disp 'To f i n d the f o u r i e r transform of given
f u n c t i o n ');
4syms x s
5F=integ(
6disp
7// >
8F1=integ ((x*
Scilab code Exa 22.4nding fourier sine transform
1//
2//
3disp 'To f i n d the f o u r i e r s i n e transform ');
4syms x s m
5//
6fs=integ(
7disp
8//
9f=integ(x*
10disp
Scilab code Exa 22.5nding fourier cosine transform
1//
2syms x s
109

111. 3disp ' Fourier c o s i n e transform ');
4f=integ(x*
,1,2);
5disp
Scilab code Exa 22.6nding fourier sine transform
1//
2syms x s a
3disp ' Fourier c o s i n e transform ');
4f=integ(
5disp
110

112. Chapter 23
Statistical Methods
Scilab code Exa 23.1Calculating cumulative frequencies of given using
iterations on matrices
1clear
2clc
3disp ' the f i r s t row of A denotes the no . of students
f a l l i n g in the marks group s t a r t i n g from (510)
. . . t i l l (4045) ')
4A(1,:) =[5 6 15 10 5 4 2 2];
5disp ' the second row denotes cumulative frequency (
l e s s than ) ')
6A(2,1) =5;
7for
8 A(2,i)=A(2,i-1)+A(1,i);
9end
10disp ' the t h i r d row denotes cumulative frequency (
more than ) ')
11A(3,1) =49;
12for
13 A(3,i)=A(3,i-1)-A(1,i-1);
14end
15disp
111

113. Scilab code Exa 23.2Calculating mean of of statistical data performing
iterations matrices
1clc
2disp ' the f i r s t row of A r e p r e s e n t s the mid v a l u e s
of weekly e a r n i n g s having i n t e r v a l of 2 in each
c l a s s=x ')
3A(1,:) =[11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
41]
4disp ' the second row denotes the no . of employees or
in other words frequency=f ' )
5A(2,:) =[3 6 10 15 24 42 75 90 79 55 36 26 19 13 9 7]
6disp ' t h i r d row denotes fx ')
7for
8 A(3,i)=A(1,i)*A(2,i);
9end
10disp ' f o u r t h row denotes u=(x25) /2 ')
11for
12 A(4,i)=(A(1,i) -25)/2
13end
14disp ' f i f t h row denotes fx ')
15for
16 A(5,i)=A(4,i)*A(2,i);
17end
18A
19b=0;
20disp 'sum of a l l elements of t h i r d row= ')
21for
22 b+=A(3,i)
23end
24disp
25f=0;
26disp 'sum of a l l elements of second row= ' )
27for
112

114. 28 f+=A(2,i)
29end
30disp
31disp ' mean= ')
32b/f
33d=0;
34disp 'sum of a l l elements of f i f t h row= ')
35for
36 d+=A(5,i)
37end
38disp ' mean by step d e v i a t i o n method= ')
3925+(2*d/f)
Scilab code Exa 23.3Analysis of statistical data performing iterations on
matrices
1clear
2clc
3disp ' the f i r s t row of A denotes the no . of students
f a l l i n g in the marks group s t a r t i n g from (510)
. . . t i l l (4045) ')
4A(1,:) =[5 6 15 10 5 4 2 2];
5disp ' the second row denotes cumulative frequency (
l e s s than ) ')
6A(2,:) =[5 11 26 36 41 45 47 49]
7disp ' the t h i r d row denotes cumulative frequency (
more than ) ')
8A(3,:) =[49 44 38 23 13 8 4 2]
9disp ' median f a l l s in the c l a s s (1520) = l +((n/2c )
h ) / f= ')
1015+((49/2 -11) *5) /15
11disp ' lower q u a r t i l e a l s o f a l l s in the c l a s s (1520)
= ')
12Q1 =15+((49/4 -11) *5) /15
13disp ' upper q u a r t i l e a l s o f a l l s in the c l a s s (2530)
113

115. = ')
14Q3 =25+((3*49/4 -36) *5)/5
15disp ' semi i n t e r q u a r t i l e range= ')
16(Q3 -Q1)/2
Scilab code Exa 23.4Analysis of statistical data
1clear
2clc
3disp ' the f i r s t row of A denotes the r o l l no . of
students form 1 to 10 and that of B denotes form
11 to 20 ')
4A(1,:) =[1 2 3 4 5 6 7 8 9 10];
5B(1,:) =[11 12 13 14 15 16 17 18 19 20];
6disp ' the second row of A annd B denotes the
co rr es po nd in g marks in p h y s i c s ')
7A(2,:) =[53 54 52 32 30 60 47 46 35 28];
8B(2,:) =[25 42 33 48 72 51 45 33 65 29];
9disp ' the t h i r d row denotes the cor re sp on di ng marks
in chemistry ')
10A(3,:) =[58 55 25 32 26 85 44 80 33 72];
11B(3,:) =[10 42 15 46 50 64 39 38 30 36];
12disp ' median marks in p h y s i c s =a r i t h m e t i c mean of 10
thand 11 th student = ' )
13(28+25) /2
14disp ' median marks in chemistry =a r i t h m e t i c mean of
10 thand 11 th student = ' )
15(72+10) /2
Scilab code Exa 23.5Finding the missing frequency of given statistical
data using given constants
1clear
114

116. 2clc
3disp ' l e t the m i s s s i n g f r e q u e n c i e s be f1and f2 ')
4disp 'sum of given f r e q u e n c i e s =12+30+65+25+18= ')
5c=12+30+65+25+18
6disp ' so , f1+f2 =229c= ')
7229-c
8disp ' median =46=40+(114.5(12+30+ f1 ) )10/65) ')
9disp ' f1 =33.5=34 ')
10f1=34
11f2=45
Scilab code Exa 23.6Calculating average speed
1clear
2clc
3syms s;
4disp ' l e t the e q i d i s t a n c e be s , then ')
5t1=s/30
6t2=s/40
7t3=s/50
8disp ' average speed=t o t a l d i s t a n c e / t o t a l time taken '
)
93*s/(t1+t2+t3)
Scilab code Exa 23.7Calculating mean and standard deviation perform-
ing iterations on matrices
1clear
2clc
3disp ' the f i r s t row denotes the s i z e of item ')
4A(1,:) =[6 7 8 9 10 11 12];
5disp ' the second row denotes the c or re sp on di ng
frequency ( f ) ')
115

117. 6A(2,:) =[3 6 9 13 8 5 4];
7disp ' the t h i r d row denotesthe c orr es po nd in g
d e v i a t i o n ( d ) ')
8A(3,:)=[-3 -2 -1 0 1 2 3];
9disp ' the f o u r t h row denotes the c or re sp ond in g fd '
)
10for
11 A(4,i)=A(2,i)*A(3,i);
12end
13disp ' the f i f t h row denotes the co rr es po nd in g fd^2 '
)
14for
15 A(5,i)=A(2,i)*(A(3,i)^2);
16end
17A
18b=0;
19disp 'sum of f o u r t h row elements= ')
20for
21 b=b+A(4,i);
22end
23disp
24c=0
25disp 'sum of f i f t h row elements= ')
26for
27 c=c+A(5,i);
28end
29disp
30d=0;
31disp 'sum of a l l f r e q u e n c i e s= ')
32for
33 d=d+A(2,i);
34end
35disp
36disp ' mean=9+b/d= ')
379+b/d
38disp ' standard d e v i a t i o n =(c /d ) ^0.5 ')
39(c/d)^0.5
116

118. Scilab code Exa 23.8Calculating mean and standard deviation perform-
ing iterations on matrices
1clc
2disp ' the f i r s t row of A r e p r e s e n t s the mid v a l u e s
of wage c l a s s e s having i n t e r v a l of 8 in each
c l a s s=x ')
3A(1,:) =[8.5 16.5 24.5 32.5 40.5 48.5 56.5 64.5 72.5]
4disp ' the second row denotes the no . of men or in
other words frequency=f ' )
5A(2,:) =[2 24 21 18 5 3 5 8 2]
6disp ' t h i r d row denotes fx ')
7for
8 A(3,i)=A(1,i)*A(2,i);
9end
10disp ' f o u r t h row denotes d=(x32.5) /8 ')
11for
12 A(4,i)=(A(1,i) -32.5)/8
13end
14disp ' f i f t h row denotes fd ')
15for
16 A(5,i)=A(4,i)*A(2,i);
17end
18disp ' s i x t h row denotes f( d ^2) ')
19for
20 A(6,i)=A(4,i)^2*A(2,i);
21end
22A
23b=0;
24disp 'sum of a l l elements of s i x t h row= ')
25for
26 b+=A(6,i)
27end
28disp
117

119. 29f=0;
30disp 'sum of a l l elements of second row= ' )
31for
32 f+=A(2,i)
33end
34disp
35disp ' mean= ')
36b/f
37d=0;
38disp 'sum of a l l elements of f i f t h row= ')
39for
40 d+=A(5,i)
41end
42disp ' mean wage= ')
4332.5+(8*d/f)
44disp ' standard d e v i a t i o n= ')
458*(b/f-(d/f)^2)
Scilab code Exa 23.9Analysis of statistical data performing iterations on
matrices
1clear
2clc
3disp ' the f i r s t row of A denotes the s c o r e s of A
and that of B denotes that of B ' )
4A(1,:) =[12 115 6 73 7 19 119 36 84 29];
5B(1,:) =[47 12 16 42 4 51 37 48 13 0];
6disp ' the second row of A annd B denotes the
co rr es po nd in g d e v i a t i o n ')
7for
8 A(2,i)= A(1,i) -51;
9 B(2,i)=B(1,i) -51;
10 end
11disp ' the t h i r d row of A and B denotes the
co rr es po nd in g d e v i a t i o n square ')
118

120. 12for
13 A(3,i)= A(2,i)^2;
14 B(3,i)=B(2,i)^2;
15end
16A
17B
18b=0;
19disp 'sum of second row elements of A=b= ' )
20for
21 b=b+A(2,i);
22 end
23 disp
24 c=0;
25disp 'sum of second row elements of B=c= ' )
26for
27 c=c+B(2,i);
28 end
29 disp
30 d=0;
31disp 'sum of t h i r d row elements of A=d= ' )
32for
33 d=d+A(3,i);
34 end
35 disp
36 e=0;
37disp 'sum of second row elements of B=e= ' )
38for
39 e=e+B(3,i);
40 end
41 disp
42 disp ' a r i t h m e t i c mean of A= ')
43 f=51+b/10
44 disp ' standard d e v i a t i o n of A= ')
45 g=(d/10-(b/10) ^2) ^0.5
46 disp ' a r i t h m e t i c mean of B= ')
47 h=51+c/10
48 disp ' standard d e v i a t i o n of A= ')
49 i=(e/10-(c/10) ^2) ^0.5
119

121. 50 disp ' c o e f f i c i e n t of v a r i a t i o n of A= ')
51 (g/f)*100
52 disp ' c o e f f i c i e n t of v a r i a t i o n of B= ')
53 (i/h)*100
Scilab code Exa 23.10Calculating mean and standard deviation of dif-
ferent statistical data when put together
1clear
2clc
3disp ' i f m i s the mean of e n t i r e data , then ')
4m =(50*113+60*120+90*115) /(50+60+90)
5disp ' i f s i s the standard d e v i a t i o n of e n t i r e data ,
then ')
6s=(((50*6^2) +(60*7^2) +(90*8^2) +(50*3^2) +(60*4^2)
+(90*1^2))/200) ^0.5
Scilab code Exa 23.12Calculating median and quartiles of given statisti-
cal data performing iterations on matrices
1clear
2clc
3disp ' the f i r s t row of A denotes the no . of persons
f a l l i n g in the weight group s t a r t i n g from
(7080) . . . t i l l (140150) ')
4A(1,:) =[12 18 35 42 50 45 20 8];
5disp ' the second row denotes cumulative frequency ' )
6A(2,1) =12;
7for
8 A(2,i)=A(2,i-1)+A(1,i);
9 end
10disp ' median f a l l s in the c l a s s (110120) = l +((n/2
c )h ) / f= ')
120

122. 11Q2 =110+(8*10) /50
12disp ' lower q u a r t i l e a l s o f a l l s in the c l a s s
(90100)= ')
13Q1 =90+(57.5 -30) *10/35
14disp ' upper q u a r t i l e a l s o f a l l s in the c l a s s
(120130)= ')
15Q3 =120+(172.5 -157) *10/45
16disp ' q u a r t i l e c o e f f i c i e n t of skewness= ')
17(Q1 +Q3 -2*Q2)/(Q3 -Q1)
Scilab code Exa 23.13Calculating coecient of correlation
1clear
2clc
3disp ' the f i r s t row of A denotes the co rr es po nd ing I
.R. of students ')
4A(1,:) =[105 104 102 101 100 99 98 96 93 92];
5disp ' the second row denotes the c or re sp on di ng
d e v i a t i o n of I .R. ')
6for
7 A(2,i)=A(1,i) -99;
8end
9disp ' the t h i r d row denotes the square of
co rr es po nd in g d e v i a t i o n of I .R. ')
10for
11 A(3,i)=A(2,i)^2;
12end
13disp ' the f o u r t h row denotes the c or re sp ond in g E.R.
of students ')
14 A(4,:) =[101 103 100 98 95 96 104 92 97 94];
15disp ' the f i f t h row denotes the co rr es po nd in g
d e v i a t i o n of E.R. ')
16for
17 A(5,i)=A(4,i) -98;
18end
121

123. 19disp ' the s i x t h row denotes the square of
co rr es po nd in g d e v i a t i o n of E.R. ')
20for
21 A(6,i)=A(5,i)^2;
22end
23disp ' the seventh row denotes the product of the two
co rr es po nd in g d e v i a t i o n s ')
24for
25 A(7,i)=A(2,i)*A(5,i);
26end
27A
28a=0;
29disp ' the sum of elements of f i r s t row=a ')
30for
31 a=a+A(1,i);
32end
33a
34b=0;
35disp ' the sum of elements of second row=b ' )
36for
37 b=b+A(2,i);
38end
39b
40c=0;
41disp ' the sum of elements of t h i r d row=c ' )
42for
43 c=c+A(3,i);
44end
45c
46d=0;
47disp ' the sum of elements of f o u r t h row=d ')
48for
49 d=d+A(4,i);
50end
51d
52e=0;
53disp ' the sum of elements of f i f t h row=e ')
54for
122

124. 55 e=e+A(5,i);
56end
57e
58f=0;
59disp ' the sum of elements of s i x t h row=d ')
60for
61 f=f+A(6,i);
62end
63f
64g=0;
65disp ' the sum of elements of seventh row=d ' )
66for
67 g=g+A(7,i);
68end
69g
70disp ' c o e f f i c i e n t of c o r r e l a t i o n= ')
71g/(c*f)^0.5
123

125. Chapter 24
Numerical Methods
Scilab code Exa 24.1nding the roots of equation
1clc
2clear
3x= ' x ');
4p=x^3-4*x-9
5disp " Finding r o o t s of t h i s equation by b i s e c t i o n
method");
6disp ' f ( 2 ) i sve and f ( 3 ) i s +ve so a root l i e s
between 2 and 3 ');
7l=2;
8m=3;
9function
10 y=x^3-4*x-9;
11endfunction
12for
13 k=1/2*(l+m);
14if
15 l=k;
16else
17 m=k;
18 end
19end
124

126. 20disp
Scilab code Exa 24.3nding the roots of equation by the method of false
statement
1//
2disp ' f ( x )=xe ^xcos ( x ) ');
3function
4 y=x*%e^(x)-
5endfunction
6
7disp 'we are r e q u i r e d to f i n d the r o o t s of f ( x ) by
the method of f a l s e p o s i t i o n ');
8disp ' f ( 0 )=ve and f ( 1 )=+ve so s root l i e between 0
and 1 ');
9disp ' f i n d i n g the r o o t s by f a l s e p o s i t i o n method ');
10
11l=0;
12m=1;
13for
14 k=l-(m-l)*f(l)/(f(m)-f(l));
15 if
16 l=k;
17 else
18 m=k;
19 end
20end
21//
22disp ' The root of the equation i s : ');
23disp
Scilab code Exa 24.4nding rea roots of equation by regula falsi method
125

127. 1//
2disp ' f ( x )=xl o g ( x )1.2 ');
3function
4 y=x*
5endfunction
6
7disp 'we are r e q u i r e d to f i n d the r o o t s of f ( x ) by
the method of f a l s e p o s i t i o n ');
8disp ' f ( 2 )=ve and f ( 3 )=+ve so s root l i e between 2
and 3 ');
9disp ' f i n d i n g the r o o t s by f a l s e p o s i t i o n method ');
10
11l=2;
12m=3;
13for
14 k=l-(m-l)*f(l)/(f(m)-f(l));
15 if
16 l=k;
17 else
18 m=k;
19 end
20end
21//
22disp ' The root of the equation i s : ');
23disp
Scilab code Exa 24.5real roots of equation by newtons method
1//
2disp ' To f i n d the r o o t s of f ( x )=3xcos ( x )1 by
newtons method ');
3disp ' f ( 0 )=ve and f ( 1 ) i s +ve so a root l i e s
between 0 and 1 ');
4l=0;
5m=1;
126

128. 6function
7 y=3*x-
8endfunction
9x0 =0.6;
10disp ' l e t us take x0 =0.6 as the root i s c l o s e r to 1 '
);
11disp "Root i s given by r=x0f ( xn ) / der ( f ( xn ) ) ");
12disp ' approximated root in each s t e p s are ');
13for
14 k=x0 -f(x0)/
15 disp
16 x0=k;
17end
Scilab code Exa 24.6real roots of equation by newtons method
1//
2clear
3clc
4disp 'To f i n d s q u a r e r o o t of 28 by newtons method l e t
x=s q r t (28) i e x^228=0 ');
5function
6 y=x^2 -28;
7endfunction
8disp ' To f i n d the r o o t s by newtons method ');
9disp ' f ( 5 )=ve and f ( 6 ) i s +ve so a root l i e s
between 5 and 6 ');
10l=5;
11m=6;
12disp ' l e t us take x0 =5.5 ');
13disp "Root i s given by rn=xn f ( xn ) / der ( f ( xn ) ) ");
14disp ' approximated root in each s t e p s are ');
15x0 =5.5;
16for
17 k=x0 -f(x0)/
127

129. 18 disp
19 x0=k;
20end
Scilab code Exa 24.7evaluating square root by newtons iterative method
1//
2clear
3clc
4disp 'To f i n d s q u a r e r o o t of 28 by newtons method l e t
x=s q r t (28) i e x^228=0 ');
5function
6 y=x^2 -28;
7endfunction
8disp ' To f i n d the r o o t s by newtons method ');
9disp ' f ( 5 )=ve and f ( 6 ) i s +ve so a root l i e s
between 5 and 6 ');
10l=5;
11m=6;
12disp ' l e t us take x0 =5.5 ');
13disp "Root i s given by rn=xn f ( xn ) / der ( f ( xn ) ) ");
14disp ' approximated root in each s t e p s are ');
15x0 =5.5;
16for
17 k=x0 -f(x0)/
18 disp
19 x0=k;
20end
Scilab code Exa 24.10solving equations by guass elimination method
1//
128

130. 2//
method
3clc
4clear
5
6disp ' S o l u t i o n of Nequation [A ] [ X]=[ r ] ')
7n= ' Enter number of Equations : ' );
8A= ' Enter Matrix [A ] : ');
9r= ' Enter Matrix [ r ] : ');
10D=A;d=r;
11
12//
13s=0;
14for
15 if
16 k=j;
17 for
18 if
19 continue
20 end
21 break
22 end
23 B=A(j,:); C=r(j);
24 A(j,:)=A(k,:); r(j)=r(k);
25 A(k,:)=B; r(k)=C;
26 end
27 for
28 L=A(i+1,j)/A(j,j);
29 A(i+1,:)=A(i+1,:)-L*A(j,:);
30 r(i+1)=r(i+1)-L*r(j);
31 end
32 s=s+1;
33end
34//
35x(n)=r(n)/A(n,n);
36for
37 sum
38 for
129

131. 39 sum
40 end
41 x(i)=(1/A(i,i))*(r(i)-
42end
43
44//
45p=
46//
47disp '@

@ ')
48disp ' Output [B ] [ x ]=[ b ] ')
49disp ' Upper r i a n g u l a r Matrix [B] = ');
50disp ' Matrix [ b ] = ');
51disp ' s o l u t i o n of l i n e a r e q u a t i o n s : ');
52disp ' s o l v e with matlab f u n c t i o n s ( f o r checking ) : ');
disp
Scilab code Exa 24.12solving equations by guass elimination method
1//
2//
method
3clc
4clear
5
6disp ' S o l u t i o n of Nequation [A ] [ X]=[ r ] ')
7n= ' Enter number of Equations : ' );
8A= ' Enter Matrix [A ] : ');
9r= ' Enter Matrix [ r ] : ');
10D=A;d=r;
11
12//
13s=0;
14for
130

132. 15 if
16 k=j;
17 for
18 if
19 continue
20 end
21 break
22 end
23 B=A(j,:); C=r(j);
24 A(j,:)=A(k,:); r(j)=r(k);
25 A(k,:)=B; r(k)=C;
26 end
27 for
28 L=A(i+1,j)/A(j,j);
29 A(i+1,:)=A(i+1,:)-L*A(j,:);
30 r(i+1)=r(i+1)-L*r(j);
31 end
32 s=s+1;
33end
34//
35x(n)=r(n)/A(n,n);
36for
37 sum
38 for
39 sum
40 end
41 x(i)=(1/A(i,i))*(r(i)-
42end
43
44//
45p=
46//
47disp '@

@ ')
48disp ' Output [B ] [ x ]=[ b ] ')
49disp ' Upper r i a n g u l a r Matrix [B] = ');
50disp ' Matrix [ b ] = ');
131

133. 51disp ' s o l u t i o n of l i n e a r e q u a t i o ns : ');
52disp ' s o l v e with matlab f u n c t i o n s ( f o r checking ) : ');
disp
Scilab code Exa 24.13solving equations by guass elimination method
1//
2//
method
3clc
4clear
5
6disp ' S o l u t i o n of Nequation [A ] [ X]=[ r ] ')
7n= ' Enter number of Equations : ' );
8A= ' Enter Matrix [A ] : ');
9r= ' Enter Matrix [ r ] : ');
10D=A;d=r;
11
12//
13s=0;
14for
15 if
16 k=j;
17 for
18 if
19 continue
20 end
21 break
22 end
23 B=A(j,:); C=r(j);
24 A(j,:)=A(k,:); r(j)=r(k);
25 A(k,:)=B; r(k)=C;
26 end
27 for
28 L=A(i+1,j)/A(j,j);
132

134. 29 A(i+1,:)=A(i+1,:)-L*A(j,:);
30 r(i+1)=r(i+1)-L*r(j);
31 end
32 s=s+1;
33end
34//
35x(n)=r(n)/A(n,n);
36for
37 sum
38 for
39 sum
40 end
41 x(i)=(1/A(i,i))*(r(i)-
42end
43
44//
45p=
46//
47disp '@

@ ')
48disp ' Output [B ] [ x ]=[ b ] ')
49disp ' Upper r i a n g u l a r Matrix [B] = ');
50disp ' Matrix [ b ] = ');
51disp ' s o l u t i o n of l i n e a r e q u a t i o n s : ');
52disp ' s o l v e with matlab f u n c t i o n s ( f o r checking ) : ');
disp
133

135. Chapter 26
Dierence Equations and Z
Transform
Scilab code Exa 26.2nding dierence equation
1//
2syms n a b yn0 yn1 yn2
3yn=a*2^n+b*(-2)^n;
4disp ' yn= ');
5disp
6n=n+1;
7yn=
8disp ' y ( n+1)=yn1= ');
9disp
10n=n+1;
11yn=
12disp ' y ( n+2)=yn2= ');
13disp
14disp ' Eliminating a b fropm t h e s e e q u a t i o n s we get :
');
15A=[yn0 1 1;yn1 2 -2;yn2 4 4]
16y=
17disp ' The r e q u i r e d d i f f e r e n c e equation : ');
18disp
134

136. 19disp '=0 ');
Scilab code Exa 26.3solving dierence equation
1//
2syms c1 c2 c3
3disp ' Cumulative f u n c t i o n i s given by E^32E^25E
+6 =0 ' );
4E= 'E ');
5f=E^3-2*E^2-5*E+6;
6r=
7disp
8disp ' There f o r the complete s o l u t i o n i s : ');
9un=c1*(r(1))^n+c2*(r(2))^n+c3*(r(3))^n;
10disp ' un= ');
11disp
Scilab code Exa 26.4solving dierence equation
1//
2syms c1 c2 c3 n
3disp ' Cumulative f u n c t i o n i s given by E^22E+1
=0 ');
4E= 'E ');
5f=E^2-2*E+1;
6r=
7disp
8disp ' There f o r the complete s o l u t i o n i s : ');
9un=(c1+c2*n)*(r(1))^n;
10disp ' un= ');
11disp
135

137. Scilab code Exa 26.6rming bonacci dierence equation
1//
2syms c1 c2 c3 n
3disp ' For Fibonacci S e r i e s yn2=yn1+yn0 ');
4disp ' so Cumulative f u n c t i o n i s given by E^2E1
=0 ');
5E= 'E ');
6f=E^2-E-1;
7r=
8disp
9disp ' There f o r the complete s o l u t i o n i s : ');
10un=(c1)*(r(1))^n+c2*(r(2))^n;
11disp ' un= ');
12disp
13disp 'Now p u t t t i n g n=1, y=0 and n=2 , y=1 we get ');
14disp ' c1=(5s q r t ( 5 ) ) /10 c2=(5+ s q r t ( 5 ) ) /10 ');
15c1=(5-
16c2 =(5+
17un=
18disp
Scilab code Exa 26.7solving dierence equation
1//
2syms c1 c2 c3 n
3disp ' Cumulative f u n c t i o n i s given by E^24E+3
=0 ');
4E= 'E ');
5f=E^2-4*E+3;
6r=
7disp
136

138. 8disp ' There f o r the complete s o l u t i o n i s = c f + pi ')
;
9cf=c1*(r(1))^n+c2*r(2)^n;
10disp 'CF= ');
11disp
12disp ' PI = 1/(E^24E+3) [ 5 ^ n ] ');
13disp ' put E=5 ');
14disp 'We get PI=5^n/8 ');
15pi=5^n/8;
16un=cf+pi;
17disp ' un= ');
18disp
Scilab code Exa 26.8solving dierence equation
1//
2syms c1 c2 c3 n
3disp ' Cumulative f u n c t i o n i s given by E^24E+4
=0 ');
4E= 'E ');
5f=E^2-4*E+4;
6r=
7disp
8disp ' There f o r the complete s o l u t i o n i s = c f + pi ')
;
9cf=(c1+c2*n)*r(1)^n;
10disp 'CF= ');
11disp
12disp ' PI = 1/(E^24E+4) [ 2 ^ n ] ');
13disp 'We get PI=n(n1) /22^(n2) ');
14pi=n*(n-1)/factorial (2) *2^(n-2);
15un=cf+pi;
16disp ' un= ');
17disp
137

139. Scilab code Exa 26.10solving dierence equation
1//
2clc
3syms c1 c2 c3 n
4disp ' Cumulative f u n c t i o n i s given by E^24 =0 ' )
;
5E= 'E ');
6f=E^2-4;
7r=
8disp
9disp ' There f o r the complete s o l u t i o n i s = c f + pi ')
;
10cf=(c1+c2*n)*r(1)^n;
11disp 'CF= ');
12disp
13//
14disp ' PI = 1/(E^24) [ n^2+n1] ');
15disp 'We get PI=n^2/37/9n17/27 ');
16pi=-n^2/3 -7/9*n -17/27;
17un=cf+pi;
18disp ' un= ');
19disp
Scilab code Exa 26.11solving dierence equation
1//
2clc
3syms c1 c2 c3 n
4disp ' Cumulative f u n c t i o n i s given by E^22E+1
=0 ');
5E= 'E ');
138

140. 6f=E^2+2*E-1;
7r=
8disp
9disp ' There f o r the complete s o l u t i o n i s = c f + pi ')
;
10cf=(c1+c2*n)*r(1)^n;
11disp 'CF= ');
12disp
13//
14disp ' PI = 1/(E1) ^ 2 [ n^22^n ] ');
15disp 'We get PI=2^n( n^28n+20 ');
16pi=2^n*(n^2-8*n+20);
17un=cf+pi;
18disp ' un= ');
19disp
Scilab code Exa 26.12solving simultanious dierence equation
1//
2clc
3disp ' s i m p l i f i e d e q u a t i o ns are : ');
4disp ' (E3)ux+vx=x . . . . . ( i ) 3ux+(E5)vx=4^x . . . . . . ( i i
) ');
5disp ' S i m p l i f y i n g we get (E^28E+12)ux=14x4^x ');
6syms c1 c2 c3 x
7disp ' Cumulative f u n c t i o n i s given by E^28E+12
=0 ');
8E= 'E ');
9f=E^2-8*E+12;
10r=
11disp
12disp ' There f o r the complete s o l u t i o n i s = c f + pi ')
;
13cf=c1*r(1)^x+c2*r(2)^x;
14disp 'CF= ');
139

141. 15disp
16//
17disp ' s o l v i n g f o r PI ');
18disp 'We get PI= ');
19pi= -4/5*x -19/25+4^x/4;
20ux=cf+pi;
21disp ' ux= ');
22disp
23disp ' Putting in ( i ) we get vx= ');
24vx=c1*2^x-3*c2*6^x -3/5*x -34/25 -4^x/4;
25disp
Scilab code Exa 26.15.2Z transform
1//
2syms n z
3y=z^(-n);
4f=symsum(y,n,0,%inf);
5disp
Scilab code Exa 26.16evaluating u2 and u3
1//
2syms z
3// 1/
4f=(2/z^2+5/z+14) /(1/z-1)^4
5u0=limit(f,z,0);
6u1=limit (1/z*(f-u0),z,0);
7u2=limit (1/z^2*(f-u0 -u1*z),z,0);
8disp ' u2= ');
9disp
10u3=limit (1/z^3*(f-u0 -u1*z-u2*z^2),z,0);
11disp ' u3= ');
140

142. 12disp
141

143. Chapter 27
Numerical Solution of Ordinary
Dierential Equations
Scilab code Exa 27.1solving ODE with picards method
1//
2syms x
3disp ' s o l u t i o n through p i c a r d s method ');
4n= ' The no of i t e r a t i o n s r e q u i r e d ');
5disp ' y ( 0 )=1 and y ( x )=x+y ');
6yo=1;
7yn=1;
8for
9 yn=yo+integ(yn+x,x,0,x);
10end
11disp ' y= ');
12disp
Scilab code Exa 27.2solving ODE with picards method
1//
142

144. 2//
3syms x
4disp ' s o l u t i o n through p i c a r d s method ');
5n= ' The no of i t e r a t i o n s r e q u i r e d ');
6disp ' y ( 0 )=1 and y ( x )=x+y ');
7yo=1;
8y=1;
9for
10
11 f=(y-x)/(y+x);
12 y=yo+integ(f,x,0,x);
13end
14disp ' y= ');
15x=0.1;
16disp
Scilab code Exa 27.5solving ODE using Eulers method
1//
2clc
3disp ' S o l u t i o n using Eulers Method ');
4disp
5n= ' Input the number of i t e r a t i o n :');
6x=0;
7y=1;
8for
9
10y1=x+y;
11y=y+0.1* y1;
12x=x+0.1;
13end
14disp ' The value of y i s :');
15disp
143

145. Scilab code Exa 27.6solving ODE using Eulers method
1//
2clc
3disp ' S o l u t i o n using Eulers Method ');
4disp
5n= ' Input the number of i t e r a t i o n :');
6x=0;
7y=1;
8for
9
10y1=(y-x)/(y+x);
11y=y+0.02* y1;
12x=x+0.1;
13disp
14end
15disp ' The value of y i s :');
16disp
Scilab code Exa 27.7solving ODE using Modied Eulers method
1//
2clc
3disp ' S o l u t i o n using Eulers Method ');
4disp
5n= ' Input the number of i t e r a t i o n :');
6x=0.1;
7m=1;
8y=1;
9yn=1;
10y1=1;
11k=1;
144

146. 12for
13
14yn=y;
15
16
17 for
18 m=(k+y1)/2;
19 yn=y+0.1*m;
20 y1=(yn+x);
21 disp
22end
23disp ' ');
24y=yn;
25m=y1;
26 yn=yn +0.1*m;
27 disp
28 x=x+0.1;
29 yn=y;
30 k=m;
31end
32disp ' The value of y i s :');
33disp
Scilab code Exa 27.8solving ODE using Modied Eulers method
1//
2clc
3disp ' S o l u t i o n using Eulers Method ');
4disp
5n= ' Input the number of i t e r a t i o n :');
6x=0.2;
7m=0.301;
8y=2;
9yn=2;
10y1=
145

147. 11k=0.301;
12for
13
14yn=y;
15
16
17 for
18 m=(k+y1)/2;
19 yn=y+0.2*m;
20 y1=
21 disp
22end
23disp ' ');
24y=yn;
25m=y1;
26 yn=yn +0.2*m;
27 disp
28 x=x+0.2;
29 yn=y;
30 k=m;
31end
32disp ' The value of y i s :');
33disp
Scilab code Exa 27.9solving ODE using Modied Eulers method
1//
2clc
3disp ' S o l u t i o n using Eulers Method ');
4disp
5n= ' Input the number of i t e r a t i o n :');
6x=0.2;
7m=1;
8y=1;
9yn=1;
146

148. 10y1=1;
11k=1;
12for
13
14yn=y;
15
16
17 for
18 m=(k+y1)/2;
19 yn=y+0.2*m;
20 y1=(
21 disp
22end
23disp ' ');
24y=yn;
25m=y1;
26 yn=yn +0.2*m;
27 disp
28 x=x+0.2;
29 yn=y;
30 k=m;
31end
32disp ' The value of y i s :');
33disp
Scilab code Exa 27.10solving ODE using runge method
1//
2disp ' Runges method ');
3function
4 y=x+y;
5endfunction
6
7x=0;
8y=1;
147

149. 9h=0.2;
10k1=h*f(x,y);
11k2=h*f(x+1/2*h,y+1/2* k1);
12kk=h*f(x+h,y+k1);
13k3=h*f(x+h,y+kk);
14k=1/6*( k1+4*k2+k3);
15disp ' the r e q u i r e d approximate value i s :');
16y=y+k;
17disp
Scilab code Exa 27.11solving ODE using runge kutta method
1//
2disp ' Runga kutta method ');
3function
4 y=x+y;
5endfunction
6
7x=0;
8y=1;
9h=0.2;
10k1=h*f(x,y);
11k2=h*f(x+1/2*h,y+1/2* k1);
12k3=h*f(x+1/2*h,y+1/2* k2);
13k4=h*f(x+h,y+k3);
14k=1/6*( k1+2*k2+2*k3+k4);
15disp ' the r e q u i r e d approximate value i s :');
16y=y+k;
17disp
Scilab code Exa 27.12solving ODE using runge kutta method
1//
148

150. 2clc
3disp ' Runga kutta method ');
4function
5 y=(y^2-x^2)/(x^2+y^2);
6endfunction
7
8x=0;
9y=1;
10h=0.2;
11k1=h*f(x,y);
12k2=h*f(x+1/2*h,y+1/2* k1);
13k3=h*f(x+1/2*h,y+1/2* k2);
14k4=h*f(x+h,y+k3);
15k=1/6*( k1+2*k2+2*k3+k4);
16disp ' the r e q u i r e d approximate value i s :');
17y=y+k;
18disp
19disp ' to f i n d y ( 0 . 4 ) put x=0.2 y=above value i e
1.196 h=0.2 ');
20x=0.2;
21h=0.2;
22k1=h*f(x,y);
23k2=h*f(x+1/2*h,y+1/2* k1);
24k3=h*f(x+1/2*h,y+1/2* k2);
25k4=h*f(x+h,y+k3);
26k=1/6*( k1+2*k2+2*k3+k4);
27disp ' the r e q u i r e d approximate value i s :');
28y=y+k;
29disp
Scilab code Exa 27.13solving ODE using runge kutta method
1//
2clc
3disp ' Runga kutta method ');
149

151. 4function
5 yy=x+y^2;
6endfunction
7
8x=0;
9y=1;
10h=0.1;
11k1=h*f(x,y);
12k2=h*f(x+1/2*h,y+1/2* k1);
13k3=h*f(x+1/2*h,y+1/2* k2);
14k4=h*f(x+h,y+k3);
15k=1/6*( k1+2*k2+2*k3+k4);
16disp ' the r e q u i r e d approximate value i s :');
17y=y+k;
18disp
19disp ' to f i n d y ( 0 . 4 ) put x=0.2 y=above value i e
1.196 h=0.2 ');
20x=0.1;
21h=0.1;
22k1=h*f(x,y);
23k2=h*f(x+1/2*h,y+1/2* k1);
24k3=h*f(x+1/2*h,y+1/2* k2);
25k4=h*f(x+h,y+k3);
26k=1/6*( k1+2*k2+2*k3+k4);
27disp ' the r e q u i r e d approximate value i s :');
28y=y+k;
29disp
Scilab code Exa 27.14solving ODE using milnes method
1//
2clc
3syms x
4yo=0;
5y=0;
150

152. 6h=0.2;
7f=x-y^2;
8y=integ(f,x,0,x);
9y1=
10disp ' y1= ');
11disp
12f=x-y^2;
13y=integ(f,x,0,x);
14y2=yo+y;
15disp ' y2= ');
16disp
17//
18 y=x-y^2;
19//
20
21y=integ(f,x,0,x);
22y3=yo+y;
23disp ' y3= ');
24disp
25disp ' determining the i n i t i a l v a l u e s f o r milnes
method using y3 ');
26disp ' x=0.0 y0 =0.0 f0=0 ');
27disp ' x=0.2 y1= ');
28x=0.2;
29disp
30y1=
31disp ' f1= ');
32f1=float(
33disp
34disp ' x=0.4 y2= ');
35x=0.4;
36disp
37disp ' f2= ');
38f2=float(
39disp
40
41disp ' x=0.6 y3= ');
42x=0.6;
151

153. 43disp
44disp ' f3= ');
45f3=float(
46disp
47//
48disp ' Using p r e d i c t o r method to f i n d y4 ');
49x=0.8;
50y4=
51disp ' y4= ');
52disp
53f4=float(
54disp ' f4= ');
55disp
56disp ' Using p r e d i c t o r method to f i n d y5 ');
57x=1.0;
58y5=
59disp
60f5=float(
61disp ' f5= ');
62disp
63disp ' Hence y ( 1 )= ');
64disp
Scilab code Exa 27.15solving ODE using runge kutta and milnes method
1//
2clc
3disp ' Runga kutta method ');
4
5function
6 yy=x*y+y^2;
7endfunction
8y0=1;
9x=0;
10y=1;
152

154. 11h=0.1;
12k1=h*f(x,y);
13k2=h*f(x+1/2*h,y+1/2* k1);
14k3=h*f(x+1/2*h,y+1/2* k2);
15k4=h*f(x+h,y+k3);
16ka =1/6*( k1+2*k2+2*k3+k4);
17disp ' the r e q u i r e d approximate value i s :');
18y1=y+ka;
19y=y+ka;
20disp
21//
22//
23
24disp ' to f i n d y ( 0 . 4 ) put x=0.2 y=above value i e
1.196 h=0.2 ');
25x=0.1;
26h=0.1;
27k1=h*f(x,y);
28k2=h*f(x+1/2*h,y+1/2* k1);
29k3=h*f(x+1/2*h,y+1/2* k2);
30k4=h*f(x+h,y+k3);
31kb =1/6*( k1+2*k2+2*k3+k4);
32disp ' the r e q u i r e d approximate value i s :');
33y2=y+kb;
34y=y+kb;
35disp
36//
37//
38
39disp ' to f i n d y ( 0 . 4 ) put x=0.2 y=above value i e
1.196 h=0.2 ');
40x=0.2;
41h=0.1;
42k1=h*f(x,y);
43k2=h*f(x+1/2*h,y+1/2* k1);
44k3=h*f(x+1/2*h,y+1/2* k2);
45k4=h*f(x+h,y+k3);
46kc =1/6*( k1+2*k2+2*k3+k4);
153

155. 47disp ' the r e q u i r e d approximate value i s :');
48y3=y+kc;
49y=y+kc;
50disp
51//
52//
53f0=f(0,y0);
54f1=f(0.1,y1);
55f2=f(0.2,y2);
56f3=f(0.3,y3);
57disp ' y0 y1 y2 y3 are r e s p e c t i v e l y : ');
58disp
59disp ' f0 f1 f2 f3 are r e s p e c t i v e l y : ');
60disp
61disp ' f i n d i n g y4 using p r e d i c t o r s milne method x=0.4
');
62h=0.1;
63y4=y0+4*h/3*(2*f1 -f2+2*f3);
64disp ' y4= ');
65disp
66disp ' f4= ');
67f4=f(0.4,y4);
68
69disp ' using c o r r e c t o r method : ');
70y4=y2+h/3*( f2+4*f3+f4);
71disp ' y4= ');
72disp
73disp ' f4= ');
74f4=f(0.4,y4);
75disp
Scilab code Exa 27.16solving ODE using adamsbashforth method
1//
2clc
154

156. 3function
4 yy=x^2*(1+y);
5endfunction
6
7y3=1
8y2 =1.233
9y1 =1.548
10y0 =1.979
11
12f3=f(1,y3)
13f2=f(1.1,y2)
14f1=f(1.2,y1)
15f0=f(1.3,y0)
16disp ' using p r e d i c t o r method ');
17h=0.1
18y11=y0+h/24*(55*f0 -59* f1 +37*f2 -9*f3)
19disp ' y11= ');
20disp
21x=1.4;
22f11=f(1.4, y11);
23disp ' using c o r r e c t o r method ');
24y11=y0+h/24*(9* f11 +19*f0 -5*f1+f2);
25disp ' y11= ');
26disp
27f11=f(1.4, y11);
28disp ' f11= ');
29disp
Scilab code Exa 27.17solving ODE using runge kutta and adams method
1//
2clc
3disp ' Runga kutta method ');
4
5function
155

157. 6 yy=x-y^2;
7endfunction
8y0=1;
9x=0;
10y=1;
11h=0.1;
12k1=h*f(x,y);
13k2=h*f(x+1/2*h,y+1/2* k1);
14k3=h*f(x+1/2*h,y+1/2* k2);
15k4=h*f(x+h,y+k3);
16ka =1/6*( k1+2*k2+2*k3+k4);
17disp ' the r e q u i r e d approximate value i s :');
18y1=y+ka;
19y=y+ka;
20disp
21//
22//
23
24disp ' to f i n d y ( 0 . 4 ) put x=0.2 y=above value i e
1.196 h=0.2 ');
25x=0.1;
26h=0.1;
27k1=h*f(x,y);
28k2=h*f(x+1/2*h,y+1/2* k1);
29k3=h*f(x+1/2*h,y+1/2* k2);
30k4=h*f(x+h,y+k3);
31kb =1/6*( k1+2*k2+2*k3+k4);
32disp ' the r e q u i r e d approximate value i s :');
33y2=y+kb;
34y=y+kb;
35disp
36//
37//
38
39disp ' to f i n d y ( 0 . 4 ) put x=0.2 y=above value i e
1.196 h=0.2 ');
40x=0.2;
41h=0.1;
156

158. 42k1=h*f(x,y);
43k2=h*f(x+1/2*h,y+1/2* k1);
44k3=h*f(x+1/2*h,y+1/2* k2);
45k4=h*f(x+h,y+k3);
46kc =1/6*( k1+2*k2+2*k3+k4);
47disp ' the r e q u i r e d approximate value i s :');
48y3=y+kc;
49y=y+kc;
50disp
51//
52//
53f0=f(0,y0);
54f1=f(0.1,y1);
55f2=f(0.2,y2);
56f3=f(0.3,y3);
57disp ' y0 y1 y2 y3 are r e s p e c t i v e l y : ');
58disp
59disp ' f0 f1 f2 f3 are r e s p e c t i v e l y : ');
60disp
61disp ' Using adams method ');
62disp ' Using the p r e d i c t o r ');
63h=0.1;
64y4=y3+h/24*(55*f3 -59* f2 +37*f1 -9*f0);
65x=0.4;
66f4=f(0.4,y4);
67disp ' y4= ');
68disp
69disp ' using c o r r e c t o r method ');
70y4=y3+h/24*(9* f4 +19*f3 -5*f2+f1);
71disp ' y4= ');
72disp
73f4=f(0.4,y4);
74disp ' f4= ');
75disp
157

159. Scilab code Exa 27.18solving simultanious ODE using picards method
1//
2clc
3disp ' Picards method ');
4x0=0;
5y0=2;
6z0=1;
7syms x
8function
9 yy=x+z;
10endfunction
11
12function
13 yy=x-y^2;
14endfunction
15disp ' f i r s t approximation ');
16y1=y0+integ(f(x,y0 ,z0),x,x0 ,x);
17disp ' y1= ');
18disp
19z1=z0+integ(g(x,y0 ,z0),x,x0 ,x);
20disp ' z1= ');
21disp
22
23disp ' second approximation ');
24y2=y0+integ(f(x,y1 ,z1),x,x0 ,x);
25disp ' y2= ');
26disp
27z2=z0+integ(g(x,y1 ,z1),x,x0 ,x);
28disp ' z2= ');
29disp
30
31disp ' t h i r d approximation ');
32y3=y0+integ(f(x,y2 ,z2),x,x0 ,x);
33disp ' y3= ');
34disp
35z3=z0+integ(g(x,y2 ,z2),x,x0 ,x);
36disp ' z3= ');
158

160. 37disp
38x=0.1;
39disp ' y ( 0 . 1 )= ');
40disp
41disp ' z ( 0 . 1 )= ');
42disp
Scilab code Exa 27.19solving ssecond ODE using runge kutta method
1//
2clc
3syms x
4function
5 yy=z;
6endfunction
7function
8 yy=x*y^2-y^2;
9endfunction
10x0=0;
11y0=1;
12z0=0;
13h=0.2;
14disp ' using k1 k2 . . f o r f and l 1 l 2 . . . f o r g runga
kutta formulae becomes ' );
15h=0.2;
16k1=h*f(x0 ,y0 ,z0);
17l1=h*g(x0 ,y0 ,z0);
18k2=h*f(x0 +1/2*h,y0 +1/2*k1 ,z0 +1/2* l1);
19l2=h*g(x0 +1/2*h,y0 +1/2*k1 ,z0 +1/2* l1);
20k3=h*f(x0 +1/2*h,y0 +1/2*k2 ,z0 +1/2* l2);
21l3=h*g(x0 +1/2*h,y0 +1/2*k2 ,z0 +1/2* l2);
22k4=h*f(x0+h,y0+k3 ,z0+l3);
23l4=h*g(x0+h,y0+k3 ,z0+l3);
24k=1/6*( k1+2*k2+2*k3+k4);
25l=1/6*( l1+2*l2+2*l3+2*l4);
159

161. 26//
27x=0.2;
28y=y0+k;
29y1=z0+l;
30disp ' y= ');
31disp
32disp ' y1= ');
33disp
34
35y
Scilab code Exa 27.20solving ODE using milnes method
1//
2clc
160

162. Chapter 28
Numerical Solution of Partial
Dierential Equations
Scilab code Exa 28.1classication of partial dierential equation
1//
2clear
3clc
4disp 'D=B^24AC ');
5disp ' i f D<0 then e l l i p t i c i f D=0 then p a r a b o l i c
i f D>0 then hyperboic ');
6disp ' ( i ) A=x ^2 ,B1y^2 D=4^2 414=0 so The
equation i s PARABOLIC ');
7disp ' ( i i ) D=4x ^2( y^21) ');
8disp ' f o rinf<x<i n f and1<y<1 D<0 ');
9disp ' So the equation i s ELLIPTIC ');
10disp ' ( i i i ) A=1+x ^2 ,B=5+2x ^2 ,C=4+x^2 ');
11disp 'D=9>0 ');
12disp ' So the equation i s HYPERBOLIC ');
Scilab code Exa 28.2solving elliptical equation
161

163. 1//
2disp ' See f i g u r e in q u e s t i o n ');
3disp 'From symmetry u7=u1 , u8=u2 , u9=u3 , u3=u1 ,
u6=u4 , u9=u7 ');
4disp ' u5 =1/4(2000+2000+1000+1000) =1500 ' );
5u5 =1500;
6disp ' u1 =1/4(0=1500+1000+2000)=1125 ');
7u1 =1125;
8disp ' u2 =1/4(1125+1125+1000+1500) =1188 ' );
9u2 =1188;
10disp ' u4 =1/4(2000+1500+1125+1125) =1438 ');
11u4 =1438;
12disp
13disp ' I t e r a t i o n s : ');
14//
')
15for
16u11 =1/4*(1000+ u2 +500+ u4);
17u22 =1/4*( u11+u1 +1000+ u5);
18u44 =1/4*(2000+ u5+u11+u1);
19u55 =1/4*( u44+u4+u22+u2);
20disp ' ');
21disp
22u1=u11;
23u2=u22;
24u4=u44;
25u5=u55;
26end
Scilab code Exa 28.3evaluating function satisfying laplace equation
1//
2clear
3clc
4disp ' See f i g u r e in q u e s t i o n ');
162

164. 5disp 'To f i n d the i n i t i a l v a l u e s of u1 u2 u3 u4 we
assume u4=0 ');
6disp ' u1 =1/4(1000+0+1000+2000)=1000 ' );
7u1 =1000;
8disp ' u2 =1/4(1000+500+1000+500)=625 ');
9u2 =625;
10disp ' u3 =1/4(2000+0+1000+500)=875 ');
11u3 =875;
12disp ' u4=1/4(875+0+625+0)=375 ');
13u4 =375;
14disp
15disp ' I t e r a t i o n s : ');
16//
')
17for
18u11 =1/4*(2000+ u2 +1000+ u3);
19u22 =1/4*( u11 +500+1000+ u4);
20u33 =1/4*(2000+ u4+u11 +500);
21u44 =1/4*( u33 +0+ u22 +0);
22disp ' ');
23disp
24u1=u11;
25u2=u22;
26u4=u44;
27u3=u33;
28end
Scilab code Exa 28.4solution of poissons equation
1//
2clear
3clc
4disp ' See f i g u r e in q u e s t i o n ');
5disp ' using numerical p o i s s o n s equation u ( i1) ( j )+u (
i +1) ( j )+u ( i ) ( j1)+u ( i ) ( j +1)=h^2 f ( ih , jh ) ');
163

165. 6disp ' Here f ( x , y )=10(x^2+y^2+10 ');
7disp ' Here f o r u1 i =1, j =2 putting in equation t h i s
g i v e s : ');
8disp ' u1=1/4(u2+u3+150 ');
9disp ' s i m i l a r l y ');
10disp ' u2=1/4(u1+u4+180 ');
11disp ' u3=1/4(u1+u4+120 ');
12disp ' u4=1/4(u2+u3+150 ');
13disp ' reducing t h e r s e e q u a t i o n s s i n c e u4=u1 ');
14disp ' 4u1u2u3150=0 ');
15disp ' u12u2+90=0 ');
16disp ' u12u3+60=0 ');
17disp ' Solvng t h e s e e q u a t i o n s by Gauss jordon method
');
18A=[4 -1 -1;1 -2 0;1 0 -2];
19r=[150; -90; -60];
20D=A;d=r;
21n=3;
22
23//
24s=0;
25for
26 if
27 k=j;
28 for
29 if
30 continue
31 end
32 break
33 end
34 B=A(j,:); C=r(j);
35 A(j,:)=A(k,:); r(j)=r(k);
36 A(k,:)=B; r(k)=C;
37 end
38 for
39 L=A(i+1,j)/A(j,j);
40 A(i+1,:)=A(i+1,:)-L*A(j,:);
41 r(i+1)=r(i+1)-L*r(j);
164

166. 42 end
43 s=s+1;
44end
45//
46x(n)=r(n)/A(n,n);
47for
48 sum
49 for
50 sum
51 end
52 x(i)=(1/A(i,i))*(r(i)-
53end
54
55//
56p=
57//
58disp '@

@ ')
59disp ' Output [B ] [ x ]=[ b ] ')
60disp ' Upper r i a n g u l a r Matrix [B] = ');
61disp ' Matrix [ b ] = ');
62disp ' s o l u t i o n of l i n e a r e q u a t i o n s : ');
Scilab code Exa 28.5solving parabolic equation
1//
2clear
3clc
4disp ' Here c^2=4 , h=1 , k=1/8 , t h e r e f o r e alpha =(c
^2)k /( h ^2) ');
5disp ' Using bendres c h m i d i t s r e c u r r e n c e r e l a t i o n i e
u ( i ) ( j +1)=tu ( i1) ( j )+tu ( i +1) ( j )+(12t )u ( i , j ) ')
;
6disp 'Now s i n c e u (0 , t )=0=u (8 , t ) t h e r e f o r e u (0 , i )=0
165

167. and u (8 , j )=0 and u ( x , 0 ) =4x1/2x^2 ');
7c=2;
8h=1;
9k=1/8;
10t=(c^2)*k/(h^2);
11A=
12
13for
14 for
15 A(1,i)=0;
16 A(9,i)=0;
17 A(i,1) =4*(i-1) -1/2*(i-1) ^2;
18
19end
20end
21//
22//
23for
24 for
25// ( 1,1) 1)
26A(i,j)=t*A(i-1,j-1)+t*A(i+1,j-1) +(1 -2*t)*A(i-1,j-1)
;
27end
28end
29for
30 j=2;
31 disp
32
33end
Scilab code Exa 28.6solving heat equation
1//
2clear
3clc
166

168. 4disp ' Here c^2=1 , h=1/3 , k=1/36 , t h e r e f o r e t =(c
^2)k /( h ^2) =1/4 ');
5disp ' So bendres c h m i d i t s r e c u r r e n c e r e l a t i o n i e u ( i
) ( j +1)=1/4(u ( i1) ( j )+u ( i +1) ( j )+2u ( i , j ) ');
6disp 'Now s i n c e u (0 , t )=0=u (1 , t ) t h e r e f o r e u (0 , i )=0
and u (1 , j )=0 and u ( x , 0 )=s i n ( %pi ) x ');
7c=1;
8h=1/3;
9k=1/36;
10t=(c^2)*k/(h^2);
11A=
12
13for
14 for
15 A(1,i)=0;
16 A(2,i)=0;
17 A(i,1)=
18
19end
20end
21//
22//
23for
24 for
25 // ( 1,1) 1) A1,
1)
26 A(i,j)=t*A(i-1,j-1)+t*A(i+1,j-1) +(1 -2*t)*A(i-1,
j-1);
27end
28end
29for
30 j=2;
31 disp
32
33end
167

169. Scilab code Exa 28.7solving wave equation
1//
2clear
3clc
4disp ' Here c ^2=16 , taking h=1 , f i n d i n g k such that
c ^2 t ^2=1 ');
5disp ' So bendres c h m i d i t s r e c u r r e n c e r e l a t i o n i e u ( i
) ( j +1)=(16 t ^2( u ( i1) ( j )+u ( i +1) ( j ) ) +2(116t ^2u ( i ,
j )u ( i ) ( j1) ');
6disp 'Now s i n c e u (0 , t )=0=u (5 , t ) t h e r e f o r e u (0 , i )=0
and u (5 , j )=0 and u ( x , 0 )=x^2(5x ) ');
7c=4;
8h=1;
9k=(h/c);
10t=k/h;
11A=
12disp ' Also from 1 s t d e r i v a t i v e ( u ( i ) ( j +1)u ( i , j1) )
/2k=g ( x ) and g ( x )=0 in t h i s case ');
13disp ' So i f j =0 t h i s g i v e s u ( i ) ( 1 ) =1/2(u ( i1) ( 0 )+u (
i +1) ( 0 ) ) ')
14for
15 for
16 A(1,i+1) =0;
17 A(6,i+1) =0;
18 A(i+1,1)=(i)^2*(5 -i);
19
20
21end
22end
23for
24 A(i+1,2) =1/2*(A(i,1)+A(i+2,1));
25
26 end
168

170. 27 for
28 for
29
30 A(i-1,j)=(c*t)^2*(A(i-2,j-1)+A(i,j-1))+2*(1 -(c*t
)^2)*A(i-1,j-1)-A(i-1,j-2);
31end
32end
33
34for
35for
36 disp
37end
38end
Scilab code Exa 28.8solving wave equation
1//
2clear
3clc
4disp ' Here c^2=4 , taking h=1 , f i n d i n g k such that
c ^2 t ^2=1 ');
5disp ' So bendres c h m i d i t s r e c u r r e n c e r e l a t i o n i e u ( i
) ( j +1)=(16 t ^2( u ( i1) ( j )+u ( i +1) ( j ) ) +2(116t ^2u ( i ,
j )u ( i ) ( j1) ');
6disp 'Now s i n c e u (0 , t )=0=u (4 , t ) t h e r e f o r e u (0 , i )=0
and u (4 , j )=0 and u ( x , 0 )=x(4x ) ');
7c=2;
8h=1;
9k=(h/c);
10t=k/h;
11A=
12disp ' Also from 1 s t d e r i v a t i v e ( u ( i ) ( j +1)u ( i , j1) )
/2k=g ( x ) and g ( x )=0 in t h i s case ');
13disp ' So i f j =0 t h i s g i v e s u ( i ) ( 1 ) =1/2(u ( i1) ( 0 )+u (
i +1) ( 0 ) ) ')
169

171. 14for
15 for
16 A(1,i+1) =0;
17 A(5,i+1) =0;
18 A(i+1,1)=(i)*(4-i);
19
20
21end
22end
23for
24 A(i+1,2) =1/2*(A(i,1)+A(i+2,1));
25
26 end
27 for
28 for
29
30 A(i-1,j)=(c*t)^2*(A(i-2,j-1)+A(i,j-1))+2*(1 -(c*t
)^2)*A(i-1,j-1)-A(i-1,j-2);
31end
32end
33
34for
35for
36 disp
37end
38end
170

172. Chapter 34
Probability and Distributions
Scilab code Exa 34.1Calculating probability
1clear
2clc
3disp ' from the p r i n c i p l e of counting , the r e q u i r e d no
. of ways are 1211109= ')
412*11*10*9
Scilab code Exa 34.2.1Calculating the number of permutations
1clear
2clc
3disp ' no . of permutations = 9 ! / ( 2 !2 !2 ! ) ')
4factorial (9)/( factorial (2)*factorial (2)*factorial (2)
)
Scilab code Exa 34.2.2Number of permutations
171

173. 1clear
2clc
3disp ' no . of permutations = 9 ! / ( 2 !2 !3 !3 ! ) ')
4factorial (9)/( factorial (2)*factorial (2)*factorial (3)
*factorial (3))
Scilab code Exa 34.3.1Calculating the number of committees
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' no . of committees=C( 6 , 3 )C( 5 , 2 )= ')
7C(6,3)*C(5,2)
Scilab code Exa 34.3.2Finding the number of committees
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' no . of committees=C( 4 , 1 )C( 5 , 2 )= ')
7C(4,1)*C(5,2)
Scilab code Exa 34.3.3Finding the number of committees
1clear
2clc
172

174. 3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' no . of committees=C( 6 , 3 )C( 4 , 2 )= ')
7C(6,3)*C(4,2)
Scilab code Exa 34.4.1Finding the probability of getting a four in a sin-
gle throw of a die
1clear
2clc
3disp ' the p r o b a b i l i t y of g e t t i n g a f o u r i s 1/6= ')
41/6
Scilab code Exa 34.4.2Finding the probability of getting an even number
in a single throw of a die
1clear
2clc
3disp ' the p r o b a b i l i t y of g e t t i n g an even no . 1/2= ')
41/2
Scilab code Exa 34.5Finding the probability of 53 sundays in a leap year
1clear
2clc
3disp ' the p r o b a b i l i t y of 53 sundays i s 2/7= ')
42/7
173

175. Scilab code Exa 34.6probability of getting a number divisible by 4 under
given conditions
1clear
2clc
3disp ' the f i v e d i g i t s can be arranged in 5 ! ways = ')
4factorial (5)
5disp ' of which 4 ! w i l l begin with 0= ')
6factorial (4)
7disp ' so , t o t a l no . of f i v e d i g i t numbers=5!4!= ')
8factorial (5)-factorial (4)
9disp ' the numbers ending in 04 ,12 ,20 ,24 ,32 ,40 w i l l
be d i v i s i b l e by 4 ')
10disp ' numbers ending in 04=3! ')
11factorial (3)
12disp ' numbers ending in 12=3! 2! ')
13factorial (3)-factorial (2)
14disp ' numbers ending in 20=3! ')
15factorial (3)
16disp ' numbers ending in 24=3! 2! ')
17factorial (3)-factorial (2)
18disp ' numbers ending in 32=3! 2! ')
19factorial (3)-factorial (2)
20disp ' numbers ending in 40=3! ')
21factorial (3)
22disp ' so , t o t a l no . of f a v o u r a b l e ways=6+4+6+4+4+6= ')
236+4+6+4+4+6
24disp ' p r o b a b i l i t y =30/96= ')
2530/96
Scilab code Exa 34.7Finding the probability
174

176. 1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' t o t a l no . of p o s s i b l e c a s e s=C( 4 0 , 4 ) ')
7C(40 ,4)
8disp ' f a v o u r a b l e outcomes=C( 2 4 , 2 )C( 1 5 , 1 )= ')
9C(24 ,2)*C(15 ,1)
10disp ' p r o b a b i l i t y= ')
11(C(24 ,2)*C(15 ,1))/C(40 ,4)
Scilab code Exa 34.8Finding the probability
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' t o t a l no . of p o s s i b l e c a s e s=C( 4 0 , 4 ) ')
7C(15 ,8)
8disp ' f a v o u r a b l e outcomes=C( 2 4 , 2 )C( 1 5 , 1 )= ')
9C(5,2)*C(10 ,6)
10disp ' p r o b a b i l i t y= ')
11(C(5,2)*C(10 ,6))/C(15 ,8)
Scilab code Exa 34.9.1Finding the probability
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
175

177. 6disp ' t o t a l no . of p o s s i b l e c a s e s=C( 9 , 3 ) ')
7C(9,3)
8disp ' f a v o u r a b l e outcomes=C( 2 , 1 )C( 3 , 1 )C( 4 , 1 )= ')
9C(2,1)*C(3,1)*C(4,1)
10disp ' p r o b a b i l i t y= ')
11(C(2,1)*C(3,1)*C(4,1))/C(9,3)
Scilab code Exa 34.9.2Finding the probability
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' t o t a l no . of p o s s i b l e c a s e s=C( 9 , 3 ) ')
7C(9,3)
8disp ' f a v o u r a b l e outcomes=C( 2 , 2 )C( 7 , 1 )+C( 3 , 2 )C
( 6 , 1 )+C( 4 , 2 )C( 5 , 1 )= ')
9C(2,2)*C(7,1)+C(3,2)*C(6,1)+C(4,2)*C(5,1)
10disp ' p r o b a b i l i t y= ')
11(C(2,2)*C(7,1)+C(3,2)*C(6,1)+C(4,2)*C(5,1))/C(9,3)
Scilab code Exa 34.9.3Finding the probability
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' t o t a l no . of p o s s i b l e c a s e s=C( 9 , 3 ) ')
7C(9,3)
8disp ' f a v o u r a b l e outcomes=C( 3 , 3 )+C( 4 , 3 )= ')
9C(3,3)+C(4,3)
176

178. 10disp ' p r o b a b i l i t y= ')
115/84
Scilab code Exa 34.13probability of drawing an ace or spade from pack
of 52 cards
1clear
2clc
3disp ' p r o b a b i l i t y of drawing an ace or spade or both
from pack of 52 cards =4/52+13/52 1/52= ')
44/52+13/52 -1/52
Scilab code Exa 34.14.1Finding the probability
1clear
2clc
3disp ' p r o b a b i l i t y of f i r s t card being a king =4/52 ')
44/52
5disp ' p r o b a b i l i t y of second card being a queen =4/52 '
)
64/52
7disp ' p r o b a b i l i t y of drawing both cards in
s u c c e s s i o n =4/524/52= ')
84/52*4/52
Scilab code Exa 34.15.1Finding the probability
1clear
2clc
177

179. 3disp ' p r o b a b i l i t y of g e t t i n g 7 in f i r s t t o s s and not
g e t t i n g i t in second t o s s =1/65/6 ')
41/6*5/6
5disp ' p r o b a b i l i t y of not g e t t i n g 7 in f i r s t t o s s and
g e t t i n g i t in second t o s s =5/61/6 ')
65/6*1/6
7disp ' r e q u i r e d p r o b a b i l i t y =1/65/6+5/61/6 ')
81/6*5/6+5/6*1/6
Scilab code Exa 34.15.2Finding the probability
1clear
2clc
3disp ' p r o b a b i l i t y of not g e t t i n g 7 in e i t h e r t o s s
=5/65/6 ')
45/6*5/6
5disp ' p r o b a b i l i t y of g e t t i n g 7 at l e a s t once
=15/65/6 ')
61 -5/6*5/6
Scilab code Exa 34.15.3Finding the probability
1clear
2clc
3disp ' p r o b a b i l i t y of g e t t i n g 7 twice =1/61/6 ')
41/6*1/6
Scilab code Exa 34.16Finding the probability
1clear
178

180. 2clc
3disp ' p r o b a b i l i t y of e n g i n e e r i n g s u b j e c t being
chooosen =(1/33/8) +(2/35/8)= ')
4(1/3*3/8) +(2/3*5/8)
Scilab code Exa 34.17Finding the probability
1clear
2clc
3disp ' p r o b a b i l i t y of white b a l l being choosen
=2/66/13+4/65/13= ')
42/6*6/13+4/6*5/13
Scilab code Exa 34.18Finding the probability
1clear
2clc
3disp " chances of winning of A=1/2+(1/2) ^2 (1/2)
+(1/2) ^4(1/2) +(1/2) ^6(1/2) +..= ' )
4(1/2) /(1(1/2) ^2)
5disp ( ' chances of winning of B=1chances of winning
of A' )
612/3
Scilab code Exa 34.19.1Finding the probability
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
179

181. 5endfunction
6disp ' t o t a l no . of p o s s i b l e outcomes=C( 1 0 , 2 )= ')
7C(10 ,2)
8disp ' no . of f a v o u r a b l e outcomes=55= ')
95*5
10disp ' p= ')
1125/49
Scilab code Exa 34.19.2Finding the probability
1clear
2clc
3disp ' t o t a l no . of p o s s i b l e outcomes=109= ')
410*9
5disp ' no . of f a v o u r a b l e outcomes=55+55= ')
65*5+5*5
7disp ' p= ')
850/90
Scilab code Exa 34.19.3Finding the probability
1clear
2clc
3disp ' t o t a l no . of p o s s i b l e outcomes=109= ')
410*10
5disp ' no . of f a v o u r a b l e outcomes=55+55= ')
65*5+5*5
7disp ' p= ')
850/100
180

182. Scilab code Exa 34.20Finding the probability
1clear
2clc
3A=1/4
4B=1/3
5AorB =1/2
6AandB=A+B-AorB
7disp ' p r o b a b i l i t y of A/B=AandB/B= ')
8AandB/B
9disp ' p r o b a b i l i t y of B/A=AandB/A= ')
10AandB/A
11disp ' p r o b a b i l i t y of AandBnot=AAandB= ')
12A-AandB
13disp ' p r o b a b i l i t y of A/Bnot=AandBnot/Bnot= ')
14(1/6) /(1 -1/3)
Scilab code Exa 34.22Finding the probability
1clear
2clc
3disp ' p r o b a b i l i t y of A h i t t i n g t a r g e t =3/5 ')
4disp ' p r o b a b i l i t y of B h i t t i n g t a r g e t =2/5 ')
5disp ' p r o b a b i l i t y of C h i t t i n g t a r g e t =3/4 ')
6disp ' p r o b a b i l i t y that two s h o t s h i t =3/52/5(13/4)
+2/53/4(13/5) +3/43/5(12/5) ')
73/5*2/5*(1 -3/4) +2/5*3/4*(1 -3/5) +3/4*3/5*(1 -2/5)
Scilab code Exa 34.23Finding the probability
1clear
2clc
181

183. 3disp ' p r o b a b i l i t y of problem not g e t t i n g s o l v e d
=1/22/33/4= ')
41/2*2/3*3/4
5disp ' p r o b a b i l i t y of problem g e t t i n g s o l v e d
=1(1/22/33/4)= ')
61 -(1/2*2/3*3/4)
Scilab code Exa 34.25nding the probability
1clc
2disp ' t o t a l frequency= i n t e g r a t e ( f , x , 0 , 2 )= ')
3n= ' x^3 ',' x ',0,1)+ ' (2x ) ^3 ',' x '
,1,2)
4disp ' u1 about o r i g i n= ')
5u1 =(1/n)*( ' ( x )( x ^3) ',' x ',0,1)+
(' ( x )((2x ) ^3) ',' x ',1,2))
6disp ' u2 about o r i g i n= ')
7u2 =(1/n)*( ' ( x ^2)( x ^3) ',' x ',0,1)+
integrate ' ( x ^2)((2x ) ^3) ',' x ',1,2))
8disp ' standard d e v i a t i o n =(u2u1 ^2) ^0.5= ')
9(u2 -u1^2) ^0.5
10disp ' mean d e v i a t i o n about the mean=(1/n )( i n t e g r a t e
(jx1j( x ^3) ,x , 0 , 1 )+i n t e g r a t e (jx1j((2x ) ^3) ,x
, 1 , 2 ` ) ')
11(1/n)*( ' (1x )( x ^3) ',' x ',0,1)+ '
( x1)((2x ) ^3) ',' x ',1,2))
Scilab code Exa 34.26nding the probability
1clear
2clc
3disp ' p r o b a b i l i t y =(0.450.03)
/(0.450.03+0.250.05+0.30.04= ')
182

184. 4(0.45*0.03) /(0.45*0.03+0.25*0.05+0.3*0.04)
Scilab code Exa 34.27nding the probability
1clear
2clc
3disp ' p r o b a b i l i t y =(1/32/63/5)
/(1/32/63/5+1/31/62/5+1/33/61/5 ')
4(1/3*2/6*3/5) /(1/3*2/6*3/5+1/3*1/6*2/5+1/3*3/6*1/5)
Scilab code Exa 34.28nding the probability
1clc
2disp ' p r o b a b i l i t y of no s u c c e s s =8/27 ')
3disp ' p r o b a b i l i t y of a s u c c e s s =1/3 ')
4disp ' p r o b a b i l i t y of one s u c c e s s =4/9 ')
5disp ' p r o b a b i l i t y of two s u c c e s s e s =2/9 ')
6disp ' p r o b a b i l i t y of t h r e e s u c c e s s e s =2/9 ')
7A=[0 1 2 3;8/27 4/9 2/9 1/27]
8disp ' mean=sum of ipi= ')
9A(1,1)*A(2,1)+A(1,2)*A(2,2)+A(1,4)*A(2,4)+A(1,3)*A
(2,3)
10disp 'sum of ipi ^2= ')
11A(1,1) ^2*A(2,1)+A(1,2) ^2*A(2,2)+A(1,4) ^2*A(2,4)+A
(1,3) ^2*A(2,3)
12disp ' v a r i a n c e =(sum of ipi ^2)1= ')
13A(1,1) ^2*A(2,1)+A(1,2) ^2*A(2,2)+A(1,4) ^2*A(2,4)+A
(1,3) ^2*A(2,3) -1
Scilab code Exa 34.29nding the probability
183

185. 1clc
2syms k
3A=[0 1 2 3 4 5 6;k 3*k 5*k 7*k 9*k 11*k 13*k]
4disp ' sumof a l l pi=1 ')
5//
6disp ' hence , ')
7k=1/49
8disp ' p ( x<4)= ')
9a=A(2,1)+A(2,2)+A(2,4)+A(2,3)
10eval
11disp
12disp ' p ( x>=5)= ')
13b=A(2,6)+A(2,7)
14eval
15disp
16disp ' p(3<x<=6)= ')
17c=A(2,5)+A(2,6)+A(2,7)
18eval
19disp
20disp ' p ( x<=2)= ')
21c=A(2,1)+A(2,2) +A(2,3)
Scilab code Exa 34.30nding the probability
1clc
2syms k
3A=[0 1 2 3 4 5 6 7;0 k 2*k 2*k 3*k k^2 2*k^2 7*k^2+k
]
4disp ' sumof a l l pi=1 ')
5//
6disp ' hence , ')
7k=1/10
8disp ' p ( x<6)= ')
9a=A(2,1)+A(2,2)+A(2,4)+A(2,3)+A(2,4)+A(2,5)+A(2,6)
10eval
184

186. 11disp
12disp ' p ( x>=6)= ')
13b=A(2,7)+A(2,8)
14eval
15disp
16disp ' p(3<x<5)= ')
17c=A(2,2)+A(2,3)+A(2,4)+A(2,5)
18eval
19disp
Scilab code Exa 34.31nding the probability
1clc
2syms x;
3f=%e^(-x)
4disp ' c l e a r l y , f>0 f o r every x in ( 1 , 2 ) and i n t e g r a t e
( f , x , 0 , %inf )= ')
5integrate '%e^(y ) ',' y ',0,%inf )
6disp ' r e q u i r e d p r o b a b i l i t y=p(1<=x<=2)=i n t e g r a t e ( f , x
, 1 , 2 )= ')
7integrate '%e^(y ) ',' y ',1,2)
8disp ' cumulative p r o b a b i l i t y f u n c t i o n f ( 2 )=i n t e g r a t e
( f , x,%inf , 2 )= ')
9integrate '%e^(y ) ',' y ',0,2)
Scilab code Exa 34.33nding the probability
1clc
2syms k;
3disp ' t o t a l p r o b a b i l i t y= i n t e g r a t e ( f , x , 0 , 6 )= ')
4p= ' kx ',' x ',0,2)
5q= ' 2k ',' x ',2,4)
6r= 'kx+6k ',' x ',4,6)
185

187. Scilab code Exa 34.34nding the probability
1clc
2A=[-3 6 9;1/6 1/2 1/3]
3disp ' f i r s t row of A d i s p l a y s the value of x ')
4disp ' the second row of x d i s p l a y s the p r o b a b i l i t y
of c or re sp on di ng to x ')
5disp 'E( x )= ')
6c=A(1,1)*A(2,1)+A(1,2) *(2 ,2)+A(1,3)*A(2,3)
7disp 'E( x )^2= ')
8b=A(1,1) ^2*A(2,1)+A(1,2) ^2*(2 ,2)+A(1,3) ^2*A(2,3)
9disp 'E(2x+1)^2=E(4x^2+4x+1) '
104*b+4*c+1
Scilab code Exa 34.35nding the probability
1clc
2disp ' t o t a l frequency= i n t e g r a t e ( f , x , 0 , 2 )= ')
3n= ' x^3 ',' x ',0,1)+ ' (2x ) ^3 ',' x '
,1,2)
4disp ' u1 about o r i g i n= ')
5u1 =(1/n)*( ' ( x )( x ^3) ',' x ',0,1)+
(' ( x )((2x ) ^3) ',' x ',1,2))
6disp ' u2 about o r i g i n= ')
7u2 =(1/n)*( ' ( x ^2)( x ^3) ',' x ',0,1)+
integrate ' ( x ^2)((2x ) ^3) ',' x ',1,2))
8disp ' standard d e v i a t i o n =(u2u1 ^2) ^0.5= ')
9(u2 -u1^2) ^0.5
10disp ' mean d e v i a t i o n about the mean=(1/n )( i n t e g r a t e
(jx1j( x ^3) ,x , 0 , 1 )+i n t e g r a t e (jx1j((2x ) ^3) ,x
, 1 , 2 ` ) ')
186

188. 11(1/n)*( ' (1x )( x ^3) ',' x ',0,1)+ '
( x1)((2x ) ^3) ',' x ',1,2))
Scilab code Exa 34.38nding the probability
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' p r o b a b i l i t y that e x a c t l y two w i l l be d e f e c t i v e
=C( 1 2 , 2 )( 0 . 1 ) ^ 2( 0 . 9 ) ^10= ')
7C(12 ,2) *(0.1) ^2*(0.9) ^10
8disp ' p r o b a b i l i t y that at l e a s t two w i l l be
d e f e c t i v e =1(C( 1 2 , 0 )( 0 . 9 ) ^12+C( 1 2 , 1 )( 0 . 1 )( 0 . 9 )
^11)= ')
91-(C(12 ,0) *(0.9) ^12+C(12 ,1) *(0.1) *(0.9) ^11)
10disp ' the p r o b a b i l i t y that none w i l l be d e f e c t i v e =C
(12 ,12)( 0 . 9 ) ^12= ')
11C(12 ,12) *(0.9) ^12
Scilab code Exa 34.39nding the probability
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' p r o b a b i l i t y of 8 heads and 4 t a i l s in 12
t r i a l s=p ( 8 )=C( 1 2 , 8 )(1/2) ^8(1/2) ^4= ')
7C(12 ,8) *(1/2) ^8*(1/2) ^4
8disp ' the expected no . of such c a s e s in 256 s e t s
=256p ( 8 ) = ')
187

189. 9256*(495/4096)
Scilab code Exa 34.40nding the probability
1clear
2clc
3function
4x=factorial(a)/( factorial(b)*factorial(a-b))
5endfunction
6disp ' p r o b a b i l i t y of a d e f e c t i v e part =2/20=0.1 ')
7disp ' p r o b a b i l i t y of a non d e f e c t i v e part =0.9 ')
8disp ' p r o b a b a i l i t y of at l e a s t t h r e e d e f e c t i v e s ina
sample = ')
91-(C(20 ,0) *(0.9) ^20+C(20 ,1) *(0.1) *(0.9) ^19+C(20 ,2)
*(0.1) ^2*(0.9) ^18')
10disp ' no . of samples having t h r e e d e f e c t i v e p a r t s
=10000.323= ')
111000*0.323
188

190. Chapter 35
Sampling and Inference
Scilab code Exa 35.1calculating the SD of given sample
1clc
2disp ' suppose the coin i s unbiased ' )
3disp ' then p r o b a b i l i t y of g e t t i n g the head in a t o s s
=1/2 ')
4disp ' then , expected no . of s u c c e s s e s=a=1/2400 ')
5a=1/2*400
6disp ' observed no . of s u c c e s s e s =216 ')
7b=216
8disp ' the e x c e s s of observed value over expected
value= ')
9b-a
10disp ' S .D. of simple sampling = ( npq ) ^0.5= c ')
11c=(400*0.5*0.5) ^0.5
12disp ' hence , z=(ba ) / c= ')
13(b-a)/c
14disp ' as z<1.96 , the h y p o t h e s i s i s accepted at 5%
l e v e l of s i g n i f i c a n c e ')
Scilab code Exa 35.2Calculating SD of sample
189

191. 1clc
2disp ' suppose the d i e i s unbiased ')
3disp ' then p r o b a b i l i t y of g e t t i n g 5 or 6 with one
d i e =1/3 ')
4disp ' then , expected no . of s u c c e s s e s=a =1/39000 ')
5a=1/3*9000
6disp ' observed no . of s u c c e s s e s =3240 ')
7b=3240
8disp ' the e x c e s s of observed value over expected
value= ')
9b-a
10disp ' S .D. of simple sampling = ( npq ) ^0.5= c ')
11c=(9000*(1/3) *(2/3))^0.5
12disp ' hence , z=(ba ) / c= ')
13(b-a)/c
14disp ' as z>2.58 , the h y p o t h e s i s has to be r e j e c t e d
at 1% l e v e l of s i g n i f i c a n c e ')
Scilab code Exa 35.3Analysis of sample
1clc
2p=206/840
3disp ' q=1p ')
4q=1-p
5n=840
6disp ' standard e r r o r of the population of f a m i l i e s
having a monthly income of r s . 250 or l e s s =(p q/n
) ^0.5= ')
7(p*q/n)^0.5
8disp ' hence taking 103/420 to be the es tim ate of
f a m i l i e s having a monthly income of r s . 250 or
l e s s , the l i m i t s are 20% and 29% approximately ')
190

192. Scilab code Exa 35.4Analysis of sample
1clear
2clc
3n1 =900
4n2 =1600
5p1 =20/100
6p2 =18.5/100
7disp ' p=(n1p1+n2p2 ) /( n1+n2 ) ')
8p=(n1*p1+n2*p2)/(n1+n2)
9disp ' q=1p ')
10q=1-p
11disp ' e=(pq(1/ n1+1/n2 ) ) ^0.5 ')
12e=(p*q*((1/ n1)+(1/ n2)))^0.5
13z=(p1 -p2)/e
14disp ' as z<1, the d i f f e r e n c e between the p r o p o r t i o n s
i s not s i g n i f i c a n t . ')
Scilab code Exa 35.5Checking whether real dierence will be hidden
1clear
2clc
3p1 =0.3
4p2 =0.25
5disp ' q1=1p1 ')
6q1=1-p1
7disp ' q2=1p2 ')
8q2=1-p2
9n1 =1200
10n2 =900
11disp ' e =(( p1q1/n1 )+(p2q2/n2 ) ) ^0.5 ')
12e=((p1*q1/n1)+(p2*q2/n2))^0.5
13z=(p1 -p2)/e
14disp ' hence , i t i s l i k e l y that r e a l d i f f e r e n c e w i l l
be hidden . ')
191

193. Scilab code Exa 35.6Checking whether given sample can be regarded as
a random sample
1clear
2clc
3disp 'm and n r e p r e s e n t s mean and number of o b j e c t s
in sample r e s p e c t i v e l y ')
4m=3.4
5n=900
6M=3.25
7d=1.61
8disp ' z=(mM) /( d /( n ^ 0 . 5 ) ')
9z=(m-M)/(d/(n^0.5))
10disp ' as z>1.96 , i t cannot be regarded as a random
sample ")
Scilab code Exa 35.9Checking whethet samples can be regarded as taken
from the same population
1clc
2disp 'm1 and n1 r e p r e s e n t s mean and no . of o b j e c t s
in sample 1 ')
3disp 'm2 and n2 r e p r e s e n t s mean and no . of o b j e c t s
in sample 2 ')
4m1 =67.5
5m2=68
6n1 =1000
7n2 =2000
8d=2.5
9disp ' on the h y p o t h e s i s that the samples are drawn
from the same population of d =2.5 ,we get ' )
192

194. 10z=(m1 -m2)/(d*((1/ n1)+(1/ n2))^0.5)
11disp ' s i n c ejzj>1 . 9 6 , thus samples cannot be
regarded as drawn from the same population ' )
Scilab code Exa 35.10calculating SE of dierence of mean hieghts
1clc
2disp 'm1, d1 and n1 denotes mean , d e v i a t i o n and no . of
o b j e c t s in f i r s t sample ')
3m1 =67.85
4d1 =2.56
5n1 =6400
6disp 'm2, d2 and n2 denotes mean , d e v i a t i o n and no . of
o b j e c t s in second sample ')
7m2 =68.55
8d2 =2.52
9n2 =1600
10disp ' S .E. of the d i f f e r e n c e of the mean h e i g h t s i s
')
11e=((d1^2/n1)+(d2^2/n2))^0.5
12m1 -m2
13disp 'jm1m2j>10e , t h i s i s h i g h l y s i g n i f i c a n t . hence
, the data i n d i c a t e s that the s a i l o r s are on the
average t a l l e r than the s o l d i e r s . ')
Scilab code Exa 35.12Mean and standard deviation of a given sample
1clear
2clc
3n=9
4disp ' f i r s t of row denotes the d i f f e r e n t v a l u e s of
sample ')
5A(1,:) =[45 47 50 52 48 47 49 53 51];
193

195. 6disp ' the second row denotes the c or re sp on di ng
d e v i a t i o n ')
7for
8 A(2,i)=A(1,i) -48;
9end
10disp ' the t h i r d row denotes the cor re sp on di ng square
of d e v i a t i o n ')
11for
12 A(3,i)=A(2,i)^2;
13end
14disp ' the sum of second row elements = ' )
15a=0;
16for
17 a=a+A(2,i);
18end
19a
20disp ' the sum of t h i r d row elements ")
21b=0;
22f o r i =1:9
23 b=b+A(3 , i ) ;
24end
25b
26disp ( 'let m be the ' )
27m=48+a/n
28disp ( 'let d be the standard deviation ' )
29d=((b/n )(a/n ) ^2) ^0.5
30t=(m47.5)(n1) ^0.5/ d
Scilab code Exa 35.13Mean and standard deviation of a given sample
1clc
2disp ' d and n r e p r e s e n t s the d e v i a t i o n and no . of
o b j e c t s in given sample ')
3n=10
4d=0.04
194

196. 5m=0.742
6M=0.700
7disp ' taking the h y p o t h e s i s that the product i s not
i n f e r i o r i . e . t h e r e i s no s i g n i f i c a n t d i f f e r e n e
between m and M')
8t=(m-M)*(n-1) ^0.5/d
9disp ' d e g r e e s of freedom= ')
10f=n-1
Scilab code Exa 34.15Standard deviation of a sample
1clear
2clc
3n=11
4disp ' the f i r s t row denotes the boy no . ')
5A(1,:) =[1 2 3 4 5 6 7 8 9 10 11];
6disp ' the second row denotes the marks in t e s t I ( x1
) ')
7A(2,:) =[23 20 19 21 18 20 18 17 23 16 19];
8disp ' the t h i r d row denotes the marks in t e s t I ( x2 )
')
9A(3,:) =[24 19 22 18 20 22 20 20 23 20 17];
10disp ' the f o u r t h row denotes the d i f f e r e n c e of marks
in two t e s t s ( d ) ')
11for
12 A(4,i)=A(3,i)-A(2,i);
13end
14disp ' the f i f t h row denotes the (d1) ')
15for
16 A(5,i)=A(4,i) -1;
17end
18disp ' the s i x t h row denotes the square of elements
of f o u r t h row ')
19for
20 A(6,i)=A(4,i)^2;
195

197. 21end
22A
23a=0;
24disp ' the sum of elements of f o u r t h row= ')
25for
26 a=a+A(4,i);
27end
28a
29b=0;
30disp ' the sum of elements of s i x t h row= ')
31for
32 b=b+A(6,i);
33end
34b
35disp ' standard d e v i a t i o n ')
36d=(b/(n-1))^0.5
37t=(1 -0)*(n)^0.5/2.24
196