ENGIN 117 Study Guide - Midterm Guide: Unit Circle, Dependent And Independent Variables, Even And Odd Functions
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1. (65) find the general real solution of the equation u (x) + u(x) = 0. Because the equation is linear and has constant coe cients, it admits the solution u = epx (constant p to be determined). By substitution, we nd that epx satis es the di erential equation if p3 = 1. (1. 1) Because (1. 1) has 3 distinct roots, the di erential equation has 3 linearly independent exponential solutions. To solve (1. 1), let p = rei , r real and non negative. =r3{cos 3 + i sin 3 } (euler formula, lect. 4, eq. 12. ) Substituting (1. 2) into (1. 1), we obtain r3e3i = 1. (1. 2) (1. 3) Taking the magnitude of both sides, and using |ei | = 1 ( real), we nd that r = 1. Consequently, e3i = 1, cos 3 + i sin 3 = 1. (1. 4a, b)
