I am a graduate student studying for a qualifying exam in Partial Differential Equations. I never took an undergraduate PDE course; as such, although I have a pretty good grasp of the theoretical aspects of the subject, my knowledge of the elementary tricks and equation manipulations is abysmal, especially basic substitutions and change-of-variables that help reduce more complex PDEs into simpler equivalents, as well as methods that involve “factoring” differential operators (ex: $\partial_{xx}-\partial_{yy}=\left(\partial_{x}+\partial_{y}\right)\left(\partial_{x}-\partial_{y}\right)$) and using them to quickly conclude what the form of the change-of-coordinates should be for the method of characteristics solution of the PDE.
Examples: $$u_{t}+\nabla^{2}u+cu = f$$ $$\nabla^{2}u-u+au_{x}+bu_{y} = 0$$
Are there any sources that I can refer to to quickly learn and memorize these kinds of petty tricks?—preferably a table or list. As it is, I am totally clueless as to how to do them, and it's deeply frustrating to be in a position where one knows why a problem should have a certain answer, but be unable to get there because of a simple obstacle such as a change of variables, or the like.
Thanks!
