I don't quite agree with your solution. You should have w = k*sqrt(sin(a)). Unless you have a typo and the original DE said (sin(a)*k)^2.
An equilibrium point is a point where the potential energy function has a local extremum. It's exactly like the top or bottom of a hill and a wheel. If you're on top of a hill (local max), you have an unstable equilibrium point. If you're at the bottom of a hill (local min), you have a stable equilibrium point. So yes, it is related to a second derivative test:
1. Form the potential energy function as a function of position y.
2. Then find its extrema. At each extremum, compute the second derivative of potential energy with respect to position y.
3. If the second derivative is positive, you've got a stable equilibrium position. If negative, then it's unstable.