Question

(y – xy2) dx - (x + x2y) dy = 0​

Answer 1

It is of the form

M dx + N dy = 0

M= y — x y^2 , N = x — x^2 y

Partial derivative of M w.r.t. ‘y' = 1 — 2xy

Partial derivative of N w.r.t. ‘x' = 1 — 2xy

Since these partial derivatives are equal, hence it's an exact differential equation.

It's solution is obtained by integrating all the terms of M w.r.t. ‘x' taking ‘y' as a constant and integrating only those terms of N w.r.t. ‘y' in which ‘x' is absent and then equating the sum to a constant.

Hence the solution is

xy — (1/2) x^2 y^2 = C