I am looking for the most general form of solutions $u:\, \mathbb{R}\times \mathbb{R} \to \mathbb{C}$ of the equation $$\frac{\partial u(x,y)}{\partial x}+ \frac{\partial u(x,y)}{\partial y}=2(x+y)u(x,y)$$
If we look for solutions of the form: $u(x,y)=f(x)g(y)$ , we find $u(x,y)= \mu e^{x^2+y^2+\lambda (x-y)} , \lambda,\mu \in \mathbb{C}$ If we look for solutions of the form: $$u(x,y)=f(x-y)g(x+y)$$ , we find $$u(x,y)= e^{x^2+y^2} F (x-y)\tag{*}$$ with $F$ an arbitrary function.
Are all solutions of the form $(*)$?
Let's change the variables: $\xi = x + y; \eta = x - y$. We have:
$$u_x = u_{\xi} + u_{\eta} $$ $$u_y = u_{\xi} - u_{\eta} $$
So our equation can be overwritten:
$$u_{\xi} = \xi u$$
Hence we got:
$$ u = f(\eta) \exp\left(\frac{\xi^2}{2}\right)$$ $$ u = f(x-y) \exp\left(\frac{(x+y)^2}{2}\right) $$
This means that the solution is unique.