MATH 267 Section P
Fall 2015
EXAM 2
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Ten points each
1. Find the solution of the initial value problem
2y 00 − 5y 0 + 3y = 0
y(0) = 2 y 0 (0) = −4
Solution:
3x
y(x) = 14ex − 12e 2
2. Find the general solution of
y 000 + 9y 0 = 3 sin 2x
Solution:
y(x) = C1 + C2 cos 3x + C3 sin 3x −
3
cos 2x
10
3. Find a particular solution of
ex
y − 2y + y =
x
00
0
using the variation of parameters method.
Solution:
y(x) = xex log x − xex
Since the second term is already a solution of the homogeneous equation it would be correct also to give the answer y(x) = xex log x.
MATH 267 Section P
Fall 2015
EXAM 2
4. State the definition of the Laplace transform and use it to compute
L{f (t)} if
(
2−t
0≤t<2
f (t) =
0
2≤t<∞
Solution:
Z
L{f (t)} =
∞
e
−st
Z
2
−st
f (t) dt =
0
e
0
e−2s + 2s − 1
(2 − t) dt =
s2
5. Solve the following system for x(t), y(t) by elimination:
x0 = 2y
y 0 = 3x − y
Solution:
2
x(t) = C1 e2t − C2 e−3t
3
y(t) = C1 e2t + C2 e−3t
6. State what is meant by a fundamental set of solutions for a second order
linear homogeneous ODE and find such a fundamental set for
3
3
y 00 + y 0 + 2 y = 0
x
x
x>0
Solution: A fundamental set is any two linearly independent solutions of the given ODE. In this case one fundamental set is
(
)
√
√
cos ( 2 ln x) sin ( 2 ln x)
,
x
x
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