Finding a periodic solution to a nonlinear ordinary differential equation is in general a difficult task. Only in a very few cases can direct methods be applied to an equation to find initial values leading to a solution of the corresponding initial value problem that is periodic. Oscillatory periodic solutions have such practical importance that great effort has gone into developing techniques to find such initial conditions. Recently a new technique was presented in this journal for finding periodic solutions having the same period as the forcing part of the equation, the so called harmonic solution. In this article, the authors give a related technique for discovering initial values leading to solutions having period a multiple of the forcing, so called subharmonic solutions. They also give some problems suitable for undergraduate research that are designed to show the broad applicability of the technique. This article concentrates on Duffing's equation as representative, but the techniques developed can certainly be applied to other equations.