Chapter 2Solution of Differential Equations

Dr. Khawaja Zafar Elahi

Separation of Variables

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Example

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1. Domain of g(x) , g(x) 0

12. Domain of , g(x) 0

g(x)

13.Domain of is domain of g(x) -{g(x)=0}

g(x)

4. Domain of sinx, cosx is (- , )

Domain

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2 2

2

2

2 2

1. ( ) 2

1 12.

( )

tan tan2.

( )

4 ln 4[1 ln ]3.

4 4

24.

x xy y x yy

y y x y x

x x

y y x y x

y y y

y x x

y x y xy x x

y y x y x y x

Partial derviavtive with respect to y

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EXAMPLE

STEP 1

STEP 2STEP 3

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Solution

STEP 4

STEP 5

STEP 6

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STEP 1

STEP 2

STEP 3

STEP 4

2

1

x yy x

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Method for solving First Order

Differential Equations

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Methods for solving First order Differential Equations

1.Separable Variable

2.Homogeneous differential Equations

3.Exact Differential Equations

4.Making Exact by Integrating Factor

5.First order Linear Differential Equation

Methods

Chapter 2

Separable Variable

x is independent variable and y is dependent variable

or

are separable forms of the differential equation

or

General solution can be solved by directly integrating both the sides

+ cWhere c is constant of integration

16DO YOU REMEMBER INTEGRATION FORMULA?

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Separation of Variables

and are separable

but is not separable.

xy xy y

y

x yy

x y

Definition A differential equation of the type y’ = f(x)g(y) is separable.

Example

x

yy

Example

Separable differential equations can often be solved with direct integration. This may lead to an equation which defines the solution implicitly rather than directly.

2 2

2 212 2

y xC y x C

ydy xdx

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EXAMPLE:

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xdxdyyy 2ln2

xdxdyyy 2ln2

cxyyyy 22 ln

xdxdyyey y 22

xdxdyyey y 22

cxeyey yy 22

211 1sinsin yyyydy

xdxdyyy

xdxdyyy

2sin2

2sin21

1

cxyyyy 2212 1sin

Note1: If we have

Integrating by parts

Note.2. If we have

Integrating by parts

Note.3. If we have

yyyydy lnln

yyy eeydyye

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Example:

222 1 xdyx y e

dx

2

2

12

1xdy x e dx

y

Separable differential equation

2

2

12

1xdy x e dx

y

2u x

2 du x dx

2

1

1udy e du

y

1

1 2tan uy C e C 21

1 2tan xy C e C 21tan xy e C

Combined constants of integration

Solution:

12

Example (cont.):

222 1 xdyx y e

dx

21tan xy e C We now have y as an implicit function of x.

We can find y as an explicit function of x by taking the tangent of both sides.

21tan tan tan xy e C

2

tan xy e C