Actually, for nonlinear ODE's of order 2 and more there is a general solution method due to Lie. It is based on reducing the order of the ODE until you can reduce it to quadrature, by consecutively applying point symmetries of the ODE. A lot of 'tricks' to solve specific ODE's are doing nothing more than applying a known symmetry to solve the equation. Finding a symmetry can be quite a tremendous task that you usually don't undertake without something like Maple.
When using Maple on this ODE however, it yields a horrible expression as a solution that is pretty much useless for all practical purposes.