Find differential equation general solution

Question

The task is to find $u_{xy}+yu_x+xu_y+xyu=0$ this equation general solutions.

First of all I wrote it as $\frac{d}{dx}(u_y+yu)+x(u_y+yu)=0$

Then marked $u_y+yu=t$ and got equation $t_x+xt=0$

And found it solution: $t=e^{-\frac{x^2}{2}}c(y)$

And I don't know how to solve this differential equation $u_y+yu=e^{-\frac{x^2}{2}}c(y)$

Answer 1

Now you have

$$ u_y+y u = g(x)\phi(y) $$

then

$$ u(x,y) = e^{-\frac{y^2}{2}}\left(\psi(x)+g(x)\int_0^y e^{\frac{\eta^2}{2}}\phi(\eta)d\eta\right) $$