This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the differential equation x[dots above] + kx[dot above] + g(x,t) = [epsilon](t) where k is a constant; g is continuous, continuously differentiable with respect to x , and is periodic of period P in the variable t; [epsilon](t) is continuous and periodic of period P, and when [partial derivative of g with respect to x] satisfies some additional boundedness conditions. This means that there exist initial values x(0) = [alpha]* and x[dot above](0) = [beta]* so that the solution to the corresponding initial value problem is periodic of period P and is unique (up to a translation of the time variable) with this property. The proof of this result is constructive, so that starting with any initial conditions x(0) = [alpha] and x[dot above](0) = [beta], a path in the phase plane can be produced, starting at ([alpha], [beta]) and terminating at ([alpha]*, [beta]*). Both the theoretical proof and a constructive proof are discussed and a "Mathematica" implementation developed which yields an algorithm in the form of a Mathematica notebook (which is posted on the webpage http://pax.st.usm.edu/downloads). The algorithm is robust and can be used on differential equations whose terms do not satisfy Li and Shen's hypotheses. The ideas used reinforce concepts from beginning courses in ordinary differential equations, linear algebra, and numerical analysis. (Contains 7 figures.)