As a hobby, I have written a basic computer algebra system. My CAS handles expressions as trees. I have advanced it to the point where it can simplify expressions symbolically (i.e., sin(pi/2) returns 1), and all expressions can be reduced to a canonical form. The CAS can also differentiate expressions.
Using this paradigm, what kinds of algorithms are there for solving equations? An equation in my model would be represented as an (=) tree with two subtrees that are the left and right expressions. I know there is no "magic bullet" for solving all equations, but are there algorithms out there that are designed to symbolically solve an equation? If there aren't, what would be the general approach? What kind of classes can equations be split into (so that I might be able to implement an algorithm for each kind)? I don't want to use naive methods and then paint myself into a metaphorical corner.