I've been working on a bit of abstract calculus which allows me to solve various PDEs in a somewhat novel fashion. It occurs to me, I can also solve nonhomogeneous PDEs by my method. For example, $$ u_{xx}+u_{yy} = x^2-y^2$$ I could probably solve. But, before I get carried away with my investigation, it occurs to me I should ask:
Is there a good exposition of the physical or mathematical significance of nonhomogeneous PDEs ? Or, if possible, can you offer one here?
I know for ODEs the significance of forcing terms is, well, force. The idea of superposition and response of linear systems to external sources is one of the more satisfying chapters in the study of ODEs. I wonder, is there some such story for PDEs? In my example, I guess I know the answer, the inhomogeneity could be viewed as charge which fills the plane with a certain density. In other words, a nonhomogeneous Laplace equation is a Poisson equation. What about other PDEs? For example, the wave equation, or heat equation? I've found the nonhomogeneous problems listed in various documents, but, my search thus far has not turned up anything which satisfies my big picture curiousity. Incidentally, I am also interested in corresponding questions for systems of PDEs, so, if you have something to say in that arena feel free.
Thanks in advance for your insight!
The first application I could think of would be geometric and/or optical modeling. Multivariate polynomials describe or approximate geometries and if you differentiate and put constraints on their level sets you can be getting (systems of) differential equations of the type you have there.
And if you can use it to describe geometries you could also use it to describe the geometries of boundary conditions used for other differential equations.