I have the following 4th-order linear PDE:
$$\dfrac{\partial^{4}\Psi}{\partial\rho^{4}}+2\phi\frac{\partial^{2}\Psi}{\partial\rho\,\partial\phi}+\frac{\partial^{2}\Psi}{\partial\phi^{2}}+\phi^{2}\frac{\partial^{2}\Psi}{\partial\rho^{2}}+\frac{\partial\Psi}{\partial\rho}=0$$
which I am aware can be factorised as
$$ \dfrac{\partial^{4}\Psi}{\partial\rho^{4}}+\left(\phi\frac{\partial}{\partial\rho}+\dfrac{\partial}{\partial\phi}\right)^{2}\Psi=0.$$
This PDE arises in a matched asymptotic expansions problem I am working through, where the outer problem involves a biharmonic operator. I was wondering whether there happens to be a change of variables which converts this equation into one that can be solved much more easily, e.g. using transform methods or similarity solutions?