Particular solution of a PDE without Laplacian

Question

I am solving the following PDE

$T_{rr}+T_{r}1/r-q^2T=αe^{-{r^2/R^2}}P$

It is a thermal diffusivity problem in polar coordinates with alpha, R, P are constants and r is the variable. Using the method of Frobenius I found the solution of the homogeneous equation and now I have to find a particular solution, but due to the conditions of the problem it does not work to use the Lapplacian.

I propose the following but I can not get anything

$T=F(r)e^{-{r^2/R^2}}P$

Ideas or suggestions

Answer 1

"Ideas or suggestions"

This is a second order linear ODE with nonconstant coefficients. You can think of it this way and ignore that it came from a PDE, spherical coordinates, physical constants, etc. It is of the form: $$ y'' + f(x)y' + g(x)y = h(x) $$ Since, you have the homogenous equation, the particular solution (at least in terms of integrals), can be found by variation of parameters. See the instructions here:

http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx