Question

Solve
$\frac{dy}{dx}=\frac{x^2+xy}{x^2+y^2}$

Answer 1

$\frac{dy}{dx}=\frac{x^2+y^2}{x^2+xy}\to \left(1\right)$

Then $\frac{dy}{dx}$ is a homogenous function

Putting $y=vx$ in equation

$\frac{dy}{dx}=V+x\frac{dv}{dx}$

Put in equation $\left(1\right)$

$v+x\frac{dv}{dx}=\frac{x^2+y^2}{x^2+y^2}$

On solving we get

$\frac{1+v}{1-v}dv=\frac{1}{x}dx$

INtegrating both sides

$\int \frac{1+v}{1-v}dv=\int \frac{1}{x}dx-2y|v-1|-v=log|x|+C$

Put $y=\frac{v}{x}$

$-2y\left(y/x\right)-1-y/x=log|x|+C$