Using Abel's Theorem to Explain Repeated Roots of the Characteristic Equation

Abstract

This document describes how one can derive the solutions to a linear constant coefficient homogeneous differential equation with repeated roots in the characteristic equation with Abel's Theorem. y (n) (t) + p1(t)y (n−1) (t) + p2(t)y (n−2) (t) + · · · + pn−1(t)y ' (t) + pn(t)y(t) = 0 (1.1) is encountered by every differential equations student. We most often study second order equations, the case when n = 2, due to Newton's laws of motion and with ease one can generalize the theory to higher order. We consider the case when the coefficient functions pi(t) are constant. Under these conditions, one makes the ansatz y = e rt which transforms (1.1) into an algebraic equation, which is more easily solved. In particular, if pi = ai, with the above ansatz, we have the characteristic equation r n + a1r n−1 + a2r n−2 + · · · + an−1r + an = 0. (1.2) By factoring out the e rt , one finds that solving an algebraic equation yields solutions to the original differential equation. Most textbooks first handle distinct real roots and later come back to discuss when r = r1 is a repeated solution to (1.2). We show an approach that relies on Abel's Theorem and the idea of reduction of order can constructively establish that the repeated root r = r1 with multiplicity m gives rise to m linearly independent solutions of the form yi(t) = t i−1 e r1t for 1 ≤ i ≤ m.