PARTIAL DIFFERENTIATION & APPLICATION .

PARTIAL DIFFERENTIATION &
APPLICATION

1. PARTIAL DIFFERENTIATION &
APPLICATION
STUDENT’S NAME: SUPRIYO DANA
CLASS ROLL NO : L004; UNIVERSITY REFNO : 34223240050226
PAPER NAME : MATHEMATICS-III ; PAPER CODE: BSC-301
DEPARTMENT OF COMPUTER SCIENCE & ENGINEERING
FUTURE INSTITUTE OF TECHNOLOGY

2. FORMATION OF PARTIAL DIFFERENTIAL EQUATIONS
GENERAL FORM
DIFFERENT INTEGRALS OF PARTIAL DIFFERNETIAL EQUATION
COMPLETE SOLUTION
PARTICULAR SOLUTION
SINGULAR SOLUTION
GENERAL SOLUTION
STANDARD TYPES OF FIRST ORDER EQUATIONS
APPLICATIONS OF PARTIAL DIFFERENTIATION
CONTENTS

3. ❑A partial differential equation is an equation involving a function of two or
more variables and some of its partial derivatives. Therefore a partial
differential equation contains one dependent variable and more than
one independent variable.
Here z will be taken as the dependent variable and x and y
the independent variable so that.
❑Partial Differential Equation can be formed either by elimination of
arbitrary constants or by the elimination of arbitrary functions from a
relation involving three or more variables.
FORMATION OF PARTIAL DIFFERENTIAL EQUATIONS

4. GENERAL FORM
❑The general form of a first order partial differential
equation is
❑where x, y are two independent variables, z is the
dependent variable and and

5. DIFFERENT INTEGRALS OF PARTIAL DIFFERNETIAL EQUATION
1)COMPLETE INTEGRAL SOLUTION
2) PARTICULAR SOLUTION
3) SINGULAR SOLUTION
4) GENERAL SOLUTION

6. COMPLETE SOLUTION
❑Let
be the Partial Differential Equation.
❑The complete integral of equation (1) is given
by
❑where a and b are two arbitrary constants

7. PARTICULAR SOLUTION
❑A solution obtained by giving the particular
values to the arbitrary constants in a
complete integral is called particular
solution.

8. SINGULAR SOLUTION
❑It is the relation between those specific
variables which involves no arbitrary
constant and is not obtainable as a
particular integral from the complete
integral.
❑ So, equation is

9. GENERAL SOLUTION
❑A relation between the variables involving two independent
functions of the given variables together with an arbitrary
function of these variables is a general solution.
❑In this given equation
assume an arbitrary relation of form b= f(a)

10. ❑So, our earlier equation becomes
❑ Now, differentiating (2) with respect to a and thus we get,
❑If the eliminator of (3) and (4) exists, then it
is known as general solution.

11. STANDARD TYPES OF FIRST ORDER EQUATIONS
❑TYPE-1 : The Partial Differential equation of
the form
has solution

12. ❑TYPE-2 : The partial differentiation equation of the form
is called Clairaut's form of partial differential equations.

13. ❑TYPE-3 : If the partial differential equations is given
by
❑Then assume that

15. ❑TYPE-4 : The partial differential equation of the given form can be
solved by assuming

16. Applications of Partial Differentiation
❑Marginal Analysis: In economics, partial differentiation is used to analyze marginal
cost, marginal revenue, and other marginal quantities.
❑Thermodynamics: Partial differentiation is applied to derive relationships in
thermodynamics, such as the Maxwell relations.
❑Heat Transfer: In engineering, partial differentiation is used in the analysis of heat
transfer and fluid dynamics.
❑Option Pricing: In finance, partial differentiation is employed in the Black-Scholes
formula for option pricing.
❑Machine Learning: In machine learning, partial derivatives are used in gradient
descent algorithms for optimizing models.

17. THANK YOU