Page 102 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 5: Differential Equations

Notes
Example 4:

x 2
x
2 log ydx dy 0
y
Here M 2 log y
x

x 2
N
y

M 2x
y y

N 2x
x y

M N
So
y y

x 2
x
Thus 2 logx ydx dy dU ( , ) 0
y
y
U
2 log y
x
x
U x 2
y y

U x 2 log y C
1
Also U x 2 logy C 2

x 2 log y a is the solution where a is a constant.

(C) Integrating Factors
Let us consider the differential equation

,
M , x y dx N x y dy 0 ...(i)
whose equation can be put into the form

U , x y a ...(ii)

where a is a constant, now from (ii)

U U
dU , x y dx dy ...(iii)
x y

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