chapter 2 solution of differential equations dr. khawaja zafar elahi separation of variables
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Chapter 2 Khawaja Zafar King saud University, Riyadh7
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
Chapter 2 Khawaja Zafar King saud University, Riyadh8
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
Chapter 2 Khawaja Zafar King saud University, Riyadh13
Method for solving First Order
Differential Equations
Chapter 2Khawaja Zafar King saud
University, Riyadh14
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
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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Chapter 2Khawaja Zafar King saud
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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:
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