The Complementary Function and the General Solution

Abstract

In a first course in differential equations the basic theme consists in obtaining explicitly the complete (i.e., general) solutions of well-known types of ordinary differential equations. Among these types one would certainly like to include the linear nth-order equation with constant coefficients not only because of its simplicity but also because of its common occurrence in engineering and other sciences. However, in asserting that the so called complementary function (CF) is indeed the general solution of the homogeneous equation, some textbooks either appear to pay insufficient attention to the significance of the statement or tend to confuse the issue and often simply point to the fact that the CF contains n arbitrary constants [see, e.g., 5, 9, 10]. Some authors who recognize this as inadequate even in a first course include in their books a statement of the basic uniqueness theorem for the associated initial value problem with or without a proof in a later chapter or an apology that the proof is beyond the scope of a first course [1, 2, 3, 4, 6, 7, 8]. Either way, the situation is not very satisfactory. One purpose of this note is to indicate a simple and elementary approach by which the problem can be overcome rather effortlessly. This is done in Part I, which is presented without skipping any of the details of calculation in order to show that it is not only complete and rigorous but is even within the reach of students who have no familiarity whatever with differential equations. Another matter this note deals with concerns the question of how to obtain a particular integral of L(y) =f where L is a linear differential operator. Here the standard methods, available after determining the CF, are (1) the annihilator method of undetermined coefficients, which succeeds if L has constant coefficients and f is such that there can be found a linear differential operator M with constant coefficients satisfying M(f) = 0, and (2) Lagrange's method of variation of parameters whose applicability is not limited as in (1). Part II describes a method somewhat different from these two that applies regardless of whether or not L has constant coefficients and irrespective of the nature of f. More importantly, this method does not depend on the predetermination of the CF. However, the applicability of this method is limited to those equations in which L can be expressed as a product of first order factors raised to appropriate powers. First I prove a simple basic lemma regarding certain indefinite integrals and then proceed directly to the theorem that gives the general solution of the homogeneous equation with constant coefficients. Only mathematical induction is used in the proofs in addition to the fundamental theorem of algebra that of course is, as always, assumed. Next I indicate an alternative and completely direct approach to the solution of the nonhomogeneous equation by repeated application of the exponential shift technique. When it applies, this method gives not merely a particular integral but in fact the general solution without any need to rely upon the fundamental uniqueness theorem for its justification.