# Differential equations with periodic driving functions

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## Abstract

The utility of the Laplace transformation (and other forms of operational mathematics) for the solution of constant coefficient, linear, differential equations, and especially those equations driven by sectionally continuous and periodic functions. Is well known and recognized. In this thesis, the Laplace transformation is used to study the solutions' of periodically driven equations. Those theorems most useful in the development of transforms of periodic functions have been stated and proven. For convenience, a table of transforms of several of the most commonly encountered periodic functions is derived. It is shown that the solution of an n[raised th] order, linear, constant coefficient, differential equation driven by a sectionally continuous function is continuous, and has n-1 continuous derivatives. It is further shown that the n[raised th] derivative of the solution of such an equation is discontinuous, and has the same discontinuities as the driving function. For equations where all of the roots of the homogeneous equation are real, the solution of the nonhomogeneous equation will have a periodic part having the same period as the driving function. For periodically driven equations, the solution is usually found to contain an infinite series involving terms of both the transient and steady-state parts of the solution. To find the steady-state portion of the solution the series must ordinarily be summed, and the solution written as a sequence, where each element of the sequence is a solution of the equation valid over some specified interval, usually a rational multiple of the period. The proof of the theorem on periodicity of solutions provides a system for finding the steady-state part of the solution directly. Finally, the above method is refined into an alternate, and somewhat simpler, procedure for finding the steady-state solution of a periodically driven differential equation.