This is the one question from my prelim that still stumps me.
Consider the nonlinear ordinary differential equation $$ u'' - u^2 + 9 = 0. $$
- Convert the equation to a system of first order equations and sketch some representative solutions of this first order system. Identify all constant solutions in your sketch.
- Find a vector field which is orthogonal to the trajectories of your first order system.
- Find a scalar function on the plane whose gradient is the vector field you found in (2).
I don't have much experience with nonlinear ODE's, so I went in circles on my exam. The constant solutions are $u = \pm 3$.
I'd like to know the general process to solve these sorts of equations, but interesting tricks and "sleights of mind" would also be great to see.