Given any nth-order homogeneous linear differential equation with constant coefficients, prove that, for any solution x and any to G R, if x(t0) = x'(to) = = .T(TC_1)(0) = 0, then x = 0 (the zero function). Hint: Use mathematical induction on n as follows. First prove the conclusion for the case n = 1. Next suppose that it is true for equations of order n 1, and consider an nth-order differential equation with auxiliary polynomial p(t). Factor p(t) = q(t)(t c), and let z = q((D))x. Show that z(to) = 0 an d z' cz 0 to conclude that z = 0. Now apply the induction hypothesis.

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Analytic Methods David Levermore Department of Mathematics University of Maryland 22 January 2012 Because the presentation of this material in lecture will diﬀer from that in the book, I felt that notes that closely follow the lecture presentation might be appreciated. Contents 1. Introduction: Classiﬁcation and Overview 1.1. Classiﬁcation 2 1.2. Course Overview 4 2. First-Order Equat