Suppose we have a multidimensional Sturm-Liouville Problem like:
$$\phi_{xx}+\phi_{yy}-x\phi_x+y\phi_y+\lambda\phi=0$$
How could we convert the above problem to a Multidimensional Sturm-Liouville Problem $$\nabla(p(x,y)\nabla\phi)+q(x,y)\phi+\lambda\sigma(x,y)\phi=0$$. Typically, what is the integrating factor to be used? Should we split the above PDE to two ODEs, finding integrating factors for each and then combine, or do we use separation of variables? Thanks beforehand.
$$ \phi_{xx}-x\phi_{x}+\phi_{yy}+y\phi_{y}+\lambda \phi = 0. $$ Multiplying by $e^{-x^2/2+y^2/2}$ leads to the desired form: $$ (e^{-x^2/2+y^2/2}\phi_{x})_{x}+(e^{-x^2/2+y^2/2}\phi_{y})_{y}+\lambda e^{-x^2/2+y^2/2}\phi = 0 \\ \nabla\cdot(e^{-x^2/2+y^2/2}\phi_{x},e^{-x^2/2+y^2/2}\phi_{y})+\lambda e^{-x^2/2+y^2/2}\phi =0 \\ \nabla\cdot(e^{-x^2/2+y^2/2}\nabla\phi)+\lambda e^{-x^2/2+y^2/2}\phi = 0. $$