It is known via the Malgrange-Ehrenpreis theorem that a PDE with constant coefficients has a fundamental solution [1]. However, the situation seems more obscure for PDE with non-constant coefficients. It seems that some related solution still exist on a case-by-case basis, see for instance [2]. I am interested in the fundamental solution to the following ODE (without boundary conditions): $$ (1+ \alpha x) u_{xx} + \alpha u_{x} + (\omega^2/\alpha^2) u = 0 $$ In other words, I am interested in the function $u^\ast(x,\xi)$ solution to $$(1+\alpha x)u^\ast_{xx}(x,\xi)+\alpha u^\ast_x(x,\xi)+(\omega^2/\alpha^2)u^\ast(x,\xi)=\delta(x-\xi)$$ where $\delta$ is the Dirac distribution.
At the moment, I am not able to find its fundamental solution if it exists.
[1] https://en.wikipedia.org/wiki/Fundamental_solution
[2] Green's function for Bessel ODE