I'm trying to find the particular solution to the PDEs below from this paper:
I only took basic undergrad ODEs so I tried solving for the particular solution by using initial conditions and couldn't find where to plug those in. I also looked up modified Bessel but I'm completely baffled. All I have right now is a hand-wavy ansatz.
Never mind, I got it. I'm just rusty at solving non-homogeneous DEs. Since the DE has a second order polynomial on the RHS (i.e. the non-homogeneous part is a second order polynomial), I will assume the particular solution is also a second order polynomial of the form $Ax^2+Bx+C$ and take its first and second derivative to plug into the DE. Find A, B, and C.