I am interested in solving the inhomogenous non-autonomous ODE
$a \left(-b u(x_2)+\frac{\partial^2 u(x_2)}{x_2^2}\right)+c x_2 \left(-b u(x_2)+\frac{\partial^2 u(x_2)}{x_2^2}\right)-d\left(\frac{\partial^4 u(x_2)}{x_2^4}-2b\frac{\partial^2 u(x_2)}{x_2^2}+b^2u(x_2)\right)=f(x_2)$
The homogeneous solution is obtained as an Airy-function. The function $f(x_2)$ is also Airy function/exponential functional expression.
I am interested in solving the ODE above in general with an arbiratry function $f(x_2)$, similar as to with Greens functions for PDEs or Duhamels principle for special ODEs.
Has anyone any knowledge in this area, and can provide my with any advice?
I would highly appreciate any help.