# Thread: Separable O.D.E question

1. ## Separable O.D.E question

Hi everyone

Need help to verify this working,thank you in advance for all your kind help & support.

$x^2\frac{dy}{dx}=y^2+2xy$(Hint:Substitute $u=\frac{y}{x}$
$\frac{dy}{dx}=\frac{y^2+2xy}{x^2}$
$u+x\frac{du}{dx}=u^2+2u$
$\frac{du}{u(u+1}=\frac{dx}{x}$
$\int(\frac{1}{u}-\frac{1}{u+1})du=\int \frac{dx}{x}$
ln|u|-ln|u+1|=ln|x|+c
$ln|\frac{\frac{y}{x}}{\frac {y}{x}+1}=ln|cx|$
$\frac {y}{x+y}=cx$
y=cx(x+y)

Thank you in advance.

2. It's good. Notice that $u=0,u=1$ are also solutions.