Download 02.5 Exact Differential Eqauation

Chapter 2
Exact Differential Equation
Exact Differential Equation
Chapter 2
1
Chapter 2
2
Exact Differential Equation
Chapter 2
Chapter 2
Exact Differential Equation
Exactness Test
N ( x, y ) 
Chapter 2
f
y
3
Exact Differential Equation
STEP:1
STEP:2
Chapter 2
4
Chapter 2
STEP:3
Exact Differential Equation
Chapter 2
Find k(y)
STEP:4
STEP:5
STEP:6
Chapter 2
5
Chapter 2
6
Exact Differential Equation
Chapter 2
Chapter 2
7
Exact Differential Equation
Chapter 2
Exact Differential Equation
Chapter 2
Example : Exact quation
Exact Differential Equation
Chapter 2
IF
M
N

,
y
x
DE is not exact.
Making Exact
Chapter 2
9
Exact Differential Equation
Chapter 2
Integrating Factor
To make DE Exact
Chapter 2
Exact Differential Equation
Integrating Factor
To make DE Exact
Chapter 2
11
Chapter 2
12
To make Exact Differential Equation
Chapter 2
Chapter 2
13
To make Exact Differential Equation
Chapter 2
Chapter 2
14
To make Exact Differential Equation
Chapter 2
Chapter 2
Exercise:
Solve the differential equation by making it exact
dy
(3xy  y )  ( x  xy )
0
dx
To make Exact Differential Equation
2
2
Integrating factor is f(x) = x
Solution is
1 2 2
x y x y c
2
3
Chapter 2
15
Chapter 2
To make Exact Differential Equation
Exercise:
Integrating factor is
Solution is
g ( y )  y 2
x2  y 2  cy
x  y  5y
2
C=5
Chapter 2
2
16