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Quiz

# quiz 3 (10).pdf

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Queen's University

Mathematics & Engineering Courses

MTHE 225

Gregory G Smith

Fall

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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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