differential-calculus-topic-1 elimination of arbitraryconstant.pdf

learn how to differentiate arbitrary constant

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DIFFERENTIAL EQUATIONS
Engr. Ricky S. Valenzuela
Philippine College of Science and Technology
Old Nalsian Road, Nalsian, Calasiao, Pangasinan Phil.2418
1st Semester, Academic Year 2024-2025
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Differential Equations
Course Description
This course is intended for all engineering students to have a firm
foundation of differential equations in preparing for their
degree-specific advanced mathematics courses. It covers first order
differential equations,n
th
order linear differential equations and
systems of first order linear differential equations. It also
introduces the concept of Laplace Transforms in solving differential
equations. The students are expected to be able to recognize
different kinds of differential equations, determine the existence
and uniqueness of solution, select the appropriate methods of
solution and interpret the obtained solution. Students are also
expected to relate differential equations to various practical
engineering and scientific problems as well as employ computer
technology in solving and verifying solutions.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Topic 1: Definitions, Elimination of Arbitrary Constants
Differential Equation
It is an equation containing differentials or derivatives of a function
of one independent variable. The following are examples of
differential equations:
dy
dx
=x+ 5
dy
dx
= cosx
d
2
y
dx
2
+k
2
y= 0
(x
2
+y
2
)dx−2xydy= 0
∂
2
V
∂x
2
+
∂
2
V
∂y
2
= 0
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Topic 1: Definitions, Elimination of Arbitrary Constants
Ordinary Differential Equation
If there is a single independent variable, the derivatives are ordinary
derivatives.
d
2
y
dx
2
+ 3
dy
dx
+ 2y= 0
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Topic 1: Definitions, Elimination of Arbitrary Constants
Partial Differential Equation
If there are two or more independent variable, the derivatives are
partial derivatives.
∂z
∂x
=z+x
∂z
∂y
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Topic 1: Definitions, Elimination of Arbitrary Constants
Order of a Differential Equation
It is the order of the highest-ordered derivative appearing in the
equation.
d
2
y
dx
2
+ 3
dy
dx
+ 2y= 0
is of the second order.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Topic 1: Definitions, Elimination of Arbitrary Constants
Degree of a Differential Equation
It is the degree of the highest ordered derivative which then occurs.
(y
′′
)
2
+ (y
′
)
3
+ 3y=x
2
is of the second degree.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Topic 1: Definitions, Elimination of Arbitrary Constants
Linear vs Nonlinear Differential Equations
Figure:
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Topic 1: Definitions, Elimination of Arbitrary Constants
Example:
Figure:Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
x
2dy
dx
+x
d
2
y
dx
2+y
2
= 1
Order:2, Degree:1, Ordinary, Linear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
x
2dy
dx
+x
d
2
y
dx
2+y
2
= 1
Order:2, Degree:1, Ordinary, Linear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
`
dy
dx
´
3
+xy
d
2
y
dx
2=ky
Order:2, Degree:1, Ordinary, Nonlinear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
`
dy
dx
´
3
+xy
d
2
y
dx
2=ky
Order:2, Degree:1, Ordinary, Nonlinear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
y
′′′
−3y” + 2y
′
+y= 0 Order:3, Degree:1, Ordinary, Linear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
y
′′′
−3y” + 2y
′
+y= 0 Order:3, Degree:1, Ordinary, Linear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
∂
2
ω
∂t
2=a
2∂
2
ω
∂x
2
Order:2, Degree:1, Partial, Nonlinear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
∂
2
ω
∂t
2=a
2∂
2
ω
∂x
2
Order:2, Degree:1, Partial, Nonlinear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
2x
∂t
∂x
+ 4y
∂t
∂y
= 0
Order:1, Degree:1, Partial, Linear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Review Problems
Problem:
Give the order and the degree of the following differential
equations. State whether ordinary or partial,also whether linear or
nonlinear.
2x
∂t
∂x
+ 4y
∂t
∂y
= 0
Order:1, Degree:1, Partial, Linear
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Definitions:
Solution of a Differential Equation
⇒is any non-derivative relationship between the variables in the
equation and which satisfies the differential equations.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Definitions:
Particular Solution
⇒it is derived from the general solution wherein the arbitrary
constant becomes absolute by means of a given initial condition.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Definitions:
General Solution
⇒it it contains an arbitrary constant.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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General Solution vs Particular Solution
Example:
Given
y
′
= 2x+ 1.
Obtain the general solution and the particular solution at an initial
conditionx= 2 andy= 1.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Elimination of Arbitrary Constants
Example 1:
Eliminate the arbitrary constant
xsiny+x
2
y=c
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Elimination of Arbitrary Constants
Example 1:
Eliminate the arbitrary constant
xsiny+x
2
y=c
Answer:(siny+ 2xy)dx+ (xcosy+x
2
)dy= 0
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Elimination of Arbitrary Constants
Example 2:
Eliminate the arbitrary constant
y=c1+c2e
3x
.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Elimination of Arbitrary Constants
Example 2:
Eliminate the arbitrary constant
y=c1+c2e
3x
.
Answer:y
′′
−3y
′
= 0
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Elimination of Arbitrary Constants
Example 3:
Eliminate the arbitrary constant
x=Asin(ωt+β);ωa parameter not to be eliminated.
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS

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Elimination of Arbitrary Constants
Example 3:
Eliminate the arbitrary constant
x=Asin(ωt+β);ωa parameter not to be eliminated.
Answer:
d
2
x
dt
2+ω
2
x= 0
Engr. Ricky S. Valenzuela DIFFERENTIAL EQUATIONS