I am wondering, is there a general method one uses to seek methods of solutions without much strenuous work and calculations, and possibly a way to get non-trivial solutions for PDE's.
A simple counter example would be:
Suppose we are given: $\dfrac{\partial u}{\partial x}-2u = 0$
And are asked to find solutions that would satisfy this equation. Well a sort of trivial solution would be to say, suppose we let $u=e^{2x}$. This would imply from the equation that, $2e^{2x}-2e^{2x}=0.$
Now we all know that this would not be the case for much tougher equations to begin with, especially non-linear equations. What I want to find out is, is there a systematic manner of sifting out the other solutions that are more non-trivial without a great deal of work. Maybe like some common trick or something. It is sort of like for ODE's of certain type, we can expect an exponential form of a solution.