L
MATH 2280-001
EXAM 2
Instructions. There are 3 problems on the exam. Put your name on tile exam.
Calculators or other electronic devices are not allowed, but you are allowed one side
of an 8.5 by 11 inch sheet of notes. Show your work for full credit.
1. (33 points) Consider the mass spring equation
4!1
+ 2’ +
=
f(t)
a. Explain what a represents, as well as the terms 2a’ and
b. Find the general solution of the homogeneous problem.
c. Find the solution satisfying the initial conditions a(O)
=
4, and x’(O)
d. Express the solution found in part c, in the form Ce_7)tcos(wit
—
=
0
a).
e. If f(t) = 1 + t + sin(t), what form would a particular solution x have )from the
method of undetermined coefficients) have?
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3. (33 points) Consider the following linear system of differential equations:
=
x
y
3
3x + p
--
a. Express the system in matrix form x’
corresponding eigenvectors of A.
b. What is the general solution (express
Ax, and find the eigenvalues, and
=
in vector form)?
c. Classify the critical point (0, 0) as a stable or unstable, node, spiral, saddle or
center.
d. Sketch the phase diagram of the system. Include the eigenvectors (if real) in your
sketch, as well as the trajectories that pass through the points (1,0), (0, 1), (—1,0),
and (0 1).
—
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at
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