I am currently studying PDE's i got an almost standard methodology for the First order PDe's mostly using the characteristic's Method.Now Im on the Second order Pde's .So i wanted to get some things clear because i haven't found any good lectures and i am studying through different books and i am trying to find a methodology for them too.So We have three main categories
$$Elliptic(La place's)-Parabolic(heat-eq)-Hyperbolic(wave-eq)$$
For the wave equation i got the standard formula.
For the other two i can use the method of Separating the Variables
So one question is. Why do we classify them if a PDE is Elliptic do we have a certain method ? I have not found one .
Also if it would be possible to suggest a methodology or personal tricks that work for you for solving Second order linear Pde's Because i think i am i little confused on how to approximate solving a PVI. say like
$$ "First-check-if-it-is-elliptic. Then.. $$
I apologize if the question is not proper for the site.It is more an asking for help question i should do with a proff. Or study more but so far studying different books led me to different things on the PDE's subject.Or suggest a book that has a certain methodology.