I am interested to solve the following PDE . $$\partial_t f(t,x,y)=-y\partial_xf(t,x,y)+(x+y)\partial_y f(t,x,y)+\alpha f(t,x,y)$$ with $f:\mathbb{R}^+\times\mathbb{R}\times \mathbb{R}\to\mathbb{R}$.
This reminds me kinetic fokker planck without the diffusion term in $y$ and under harmonic potential in both $x$ and $y$.
I tried with characterist method but I think I have too many variables and I was not able to conclude.
How can I solve it (if possible?). Is there a method I am not aware of?
Hint. Let $f(x,y,t)=u(\xi,\eta,t)$, where $\xi:=x^2+y^2+xy$ and $\eta:=\frac{y}{x}$. Show that $$ -yf_x+(x+y)f_y=\xi u_{\xi}+(\eta^2+\eta+1)u_{\eta}. \tag{1} $$ Expressed in terms of the new coordinates, the PDE becomes $$ u_t=\xi u_{\xi}+(\eta^2+\eta+1)u_{\eta}+\alpha u, \tag{2} $$ for which the Lagrange-Charpit equations are separable.