Exam II Review
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Qualitative Behavior
Consider the following 2nd order ODE’s
(a) y 00 + 2y 0 + y = 0
√
(b) 3y 00 + 2y 0 + y = 0
(c) y 00 + 1.1y = cos(t)
(d) y 00 + 4y = sin(2t + π/2)
Without solving, do the following for each equation given above: name the
phenomena exhibited by the equation and give justification. Finally, sketch
the qualitative behavior of the solution.
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Solution Techniques
For each of the equations listed below, state which method is the most
appropriate solution technique (give justification). Then, solve the equation.
(a) y 00 − y 0 − 2y = et ,
y(0) = y 0 (0) = 1
R
(b) y 00 + y = sec2 (x), Hint: sec(x)dx = ln | sec(x) + tan(x)| + c
(c) y 00 + y = 1 + U t − π2 [sin(t) − 1] , y(0) = 1, y 0 (0) = 0
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Spring-Mass System
A force of 400 Newtons stretches a spring 2 meters. A mass of 50 kilograms
is attached to the end of the spring and it is initially released from the
equilibrium position with an upward velocity of 10 m/s. Additionally, the
mass is driven by an external force equal to f (t) = sin(2t).
(a) Write the corresponding IVP.
(b) Solve the IVP using any appropriate method. Name the behavior of
the solution.
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Laplace Transform
(a) For the following equation, find the Laplace transform using only the
definition:
f (t) = et cos(t)
(b) Find the Laplace transforms of the following functions using the table:
(i) g(t) = (t + 1)3
0
t < π/2
(ii) h(t) =
cos(3(t − π/2)) t ≥ π/2
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Inverse Laplace Transform
Find the inverse Laplace transform of the following functions:
(a) Y (s) =
2s+5
s2 +6s+34
(b) Q(s) =
e−s (2s−1)
s2 (s+1)2
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One More Thing
There are a few topics that are not included on this review, but that doesn’t
mean they won’t be on the exam. You are still responsible for all material
covered in lecture!
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