# Math Help - Checking differential equation

1. ## Checking differential equation

Can someone check my working?

Solve $xy'=2x^2y+y\ln y$ by letting $\ln y=v$

$\ln y=v\Rightarrow y=e^v$

$\dfrac{dy}{dx}=\dfrac{dy}{dv}\cdot\dfrac{dv}{dx}$

$\dfrac{dy}{dx}=e^v \dfrac{dv}{dx}$

$xe^v \dfrac{dv}{dx}=2x^2e^v+ve^v$

Divide by $e^v$

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

$\dfrac{dv}{dx}=2x+\dfrac{v}{x}$

$\dfrac{dv}{dx}-\dfrac{v}{x}=2x$

$I=e^{\int{-\frac{1}{x}} \ dx}=\dfrac{1}{x}$

$\dfrac{1}{x} \dfrac{dv}{dx}-\dfrac{v}{x^2}=2$

$\left[\dfrac{1}{x}v\right]'=2$

$\dfrac{1}{x}v=2x+C$

$v=2x^2+Cx$

$\ln y =2x^2+Cx$

$\Rightarrow y=e^{2x^2+Cx}$

2. It all looks good to me...