Before I delve into this , I just wanted to know the basic approach. Do we look for symmetries because it gives us a systematic way to find coordinate changes that change the differential equation into a separable one? Thanks jf
In general, you look for symmetries as a means of letting you assume particular forms of the solutions. For example, if you have a partial differential equation in three dimension that displays full rotational symmetry (including the boundary conditions!) then your solution cannot depend on the angular variables and you can reduce the problem to an ordinary differential equation for the radial solution.
Do you mean applying Lie point symmetries to find general solutions to differential equations? You can apply it as in your first post, and look for coordinate changes that will solve the (usually ordinary) differential equation. You can also use it for much more than that. Symmetries cannot be mapped into each other. A nice book to start with if you want to learn more is the introduction book (symmetry methods for differential equations) from Peter Hydon. Or if you are looking for algorithms that are systematically searching for symmetries, there are many papers from Cheb-terrab et al, who worked a lot on the ode solvers in Maple.