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Queen's University
Mathematics & Engineering Courses
MTHE 225
Gregory G Smith

Quiz 3a Solutions 1. This question concerns second-order linear homogeneous equations. Show your work. ′′ ′ (a) Find two linearly independent solutions of y + 6y + 9y = 0. The characteristic equation is r +6r +9 = 0, which factors as (r +3) = 0, 2 so there is a double root r = −3. Thus two linearly independent solutions are given by −3x −3x y1= e , y2= xe (b) Find the general solution of y + 4y = 0. The characteristic equation is r + 4 = 0, with complex conjugate roots r = ±2i. Thus two linearly independent solutions are given by y1= cos2x, y2= sin2x and the general solution is y = c 1 1 c y2 2 = c 1os2x + c si22x 2. The solution of x + 4x = 0, with initial conditions x(0) = 1, x (0) = −2, is x(t) = cosω t0− sinω t0 where I haven’t told you what ω is, but we
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