Page 103 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Thus equation (i) is exact if
U
M ,
x

U
N
y

2
U M N
or ...(iv)
x y y x
On the other hand if (iv) is not satisfied then we can multiply equation (i) by a function f(x, y)
such that

U
f , x y M M '
x
U ...(v)
f , x y N N '
y

2 U M ' N '
and so ...(vi)
x y y x
and f(x, y) M(x, y) dx + f(x, y) N(x, y) dy = 0 ...(vii)
is an exact equation. Here f(x, y) is known as integrating factor.

Example 5: Solve the differential equation by suitable integrating factor

2
xdy ydx x dx 0 ...(i)
Solution:

Here x 2 y dx xdy 0

so M x 2 y and N x

M N
Now 1, 1
y x
M N
so
y x
1
Now multiplying equation (i) by 2 , so that
x

2
xdy ydx x dx 0
x 2 x 2 ...(ii)

y
or d dx 0
x

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