$$u_{xy}=2u_{x} \implies u_{y}=2u+\varphi_{y}(y) \implies (e^{-2y}u)_{y}=e^{-2y}\varphi_{y}(y) \implies u=\varphi(y)+2e^{2y}(\int^{y}\varphi(s)e^{-2s}\,ds+\psi(x))$$

But it is really an ODE. If I have mistaken then please inform me. I have used intergration by part and standard ODE technique.

I made a lost of mistakes when I first post these, including: forget the arbitrary function after last integration wrt y; messed up the order of multiplication.

Edit:

Really such a mess is no better than something wrong. Sub in $v=u_{x}$ immediately $v=\varphi_{x}(x)e^{2y}\implies u=\psi(y)+\varphi(x)e^{2y}$.