Q10. (x-y) dy/dx =x + 2y

Here is the answer to your question.

(x - y) \frac{d y}{d x} = x + 2 y \Rightarrow \frac{d y}{d x} = \frac{x + 2 y}{x - y} F (x, y) = \frac{x + 2 y}{x - y} F (A x, A y) = \frac{A x + 2 A y}{A x - A y} = \frac{A (x + 2 y)}{A (x - y)} = \frac{x + 2 y}{x - y} = F (x, y) H e n c e, t h e e q u a t i o n i s h o m o g e n o u s .

Integrating on both the sides, we get

This is the solution of the given differential equation.

Cheers!