I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't answerable in the current state of affairs. My professor basically tells me to take an advanced course in real analysis for a rigorous treatment, but that's a long way away for me (at least two years), I was hoping that someone could answer this question here.
In the course, we have only looked at three equations and some minor generalizations on them - the heat, wave, and potential equations. I understand these to be characteristic of larger classes of PDEs, but I don't know anything at all about them except sometimes I can separate variables. For each of them, the method has been identical. Separate variables, solve two (or sometimes even three) ODEs. Then by superposition, sum them up. Determine coefficients with Fourier sums or integrals. Wash, rinse, repeat.
The question is this: How do I know that that is all the solutions? There is no existence/uniqueness theorem for PDEs. How can I know that without some more advanced technique for solving PDEs that I couldn't find others? Does it follow from existence/uniqueness of ODEs? What about for those larger classes of PDEs?