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The solutions manual for test 2 of math 285 spring 2003, focusing on differential equations and oscillations. It includes problems on finding complementary and particular solutions, undamped and damped forced oscillations, resonance, boundary value problems, and eigenvalue problems. Students are expected to explain their answers and are not allowed to use notes, books, calculators, or computers.
Typology: Exams
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Total points: 100. Do all questions. Explain all answers. No notes, books, calculators or computers.
y′′^ + 9y = x cos 3x.
yc =
yp =
x′′(t) + ω^20 x(t) = sin(2t),
by Undetermined Coefficients.
(b) Roughly sketch a typical solution x(t) for large t-values, for ω 0 = 2, 3.
mx′′(t) + x′(t) + x(t) = 0.
Find all m-values such that the amplitude of oscillation gets reduced by at least 80%, during each unit of time. (You may assume m > 1 /4.)
mx′′(t) + cx′(t) + kx(t) = F 0 cos(ωt),
where m, c, k are positive constants.
(a) Write down the form of the steady periodic response to the forcing. (You do not have to evaluate any of the coefficients.)
(b) Take x(0) = 1000, x′(0) = 2000 and ω = 2. Roughly sketch the shape of the solution x(t) for large t, as best you can. Explain.
Formulas Here are some formulas you might be able to use on the test:
y = yc + yp
ω 0 =
k m
, p =
c 2 m
, ω 1 =
ω^20 − p^2
e(a±ib)x^ = eax(cos bx ± i sin bx)
y = −y 1
y 2 f W
dx + y 2
y 1 f W
dx
W = y 1 y′ 2 − y′ 1 y 2
