Question

Solve the following differential equation: $x^2\frac{dy}{dx}=2xy+y^2$.

Answer 1

Given,

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

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

$y=vx\to \frac{dy}{dx}=x\frac{dv}{dx}+v$

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

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

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

$\log \left(\frac{v}{v+1}\right)=\log x+\log c$

$\log \left(\frac{y}{x+1}\right)=\log x+\log c$

$\frac{y}{x+1}=cx$