Hey euphony.
Those DE's always have an exact solution, so can you not just classify the solution to get the answer?
Is there a way I can determine whether or not the solutions to a (homogeneous second-order linear) differential equation are periodic just by looking at the phase plane portrait? (Will periodic solutions always be elliptical? Can spirals be periodic?)
If a function is periodic, f(t+ h)= f(t), then so are all derivatives of f. That means that we have exactly the same point on the phase plane for different t. The curve is NOT necessarily a true "ellipse" but it must be a closed curve. Spirals are not closed curves so cannot represent a periodic solution.