Page 101 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
Then equation (i) is known as exact equation. Here , are partial derivative w.r.t. x and y
X y
respectively. Let us introduce U(x, y) such that
M(x, y) dx + N(x, y) dy = dU(x, y)= 0 ...(iii)
U U
also dU(x, y) = dx dy ...(iv)
x y

From (iii) and (iv) we have
U ( , )
y
x
M ( , )
x
y
x
y
U ( , )
x
N ( , )
x
y
y
2
U ( , ) M N
x
y
Now
y x y x
Hence we have
dU (x, y) = 0
or U (x, y) = a (a being a constant) is a solution
Example 3: Solve
2xy 1 dx x 2 4y dy 0
Consider a function U(x, y) such that

dU= 2xy 1 dx x 2 4y dy 0

U U
= dx dy
x y

M N
2
So 2xy +1 = M, x 4y = N;
y x
U
2xy 1 ...(i)
x
U 2
x 4y ...(ii)
y
2
y
From (i) U = x y a ( ) b constant
4y 2
2
From (ii) U = x y c constant
2
2
Comparing we have a(y) = 2y and c = b so
2
U = x y 2y a

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