Math 2243 â Spring 2006 Final Exam Answer Key
Math 2243 â Spring 2006 Final Exam Answer Key
Math 2243 â Spring 2006 Final Exam Answer Key
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<strong>Math</strong> <strong>2243</strong> – <strong>Spring</strong> <strong>2006</strong> <strong>Final</strong> <strong>Exam</strong><br />
<strong>Answer</strong> <strong>Key</strong><br />
1. the Wronskian is W = -24 ≠ 0 , so the functions are linearly independent<br />
2.<br />
y = t + C t 2<br />
3.<br />
€<br />
a)<br />
u ' 1 = u 2<br />
u ' 2 = − 8 u 1 + 6 u 2<br />
b)<br />
y '' − 8 y ' + 14 y = 0<br />
4.<br />
€<br />
y = −<br />
1<br />
t 2 − 3t + 6<br />
€<br />
€<br />
5.<br />
T (t ) = 90 − 15 ⋅<br />
⎛⎛<br />
⎜⎜ 2 ⎞⎞ ⎟⎟ t or 90 − 15 ⋅ e − 0.4055 t º F.<br />
⎝⎝ 3⎠⎠<br />
€<br />
6. for λ = −1 : eigenvectors are<br />
⎡⎡ 1 ⎤⎤<br />
⎢⎢ ⎥⎥<br />
⎢⎢<br />
−1<br />
⎥⎥<br />
⎣⎣ ⎢⎢ 0 ⎦⎦ ⎥⎥<br />
and<br />
⎡⎡ 1 ⎤⎤<br />
⎢⎢ ⎥⎥<br />
⎢⎢<br />
0<br />
⎥⎥<br />
⎣⎣ ⎢⎢ −1⎦⎦<br />
⎥⎥<br />
[two-dimensional eigenspace] ;<br />
for λ = 2 : eigenvector € is<br />
⎡⎡ 1⎤⎤<br />
⎢⎢ ⎥⎥<br />
⎢⎢<br />
1<br />
⎥⎥<br />
⎣⎣ ⎢⎢ 1⎦⎦<br />
⎥⎥<br />
[one-dimensional eigenspace] ;<br />
7. kernel:<br />
s<br />
⎡⎡ 1 ⎤⎤<br />
⎢⎢ ⎥⎥<br />
⎢⎢<br />
−1<br />
⎥⎥<br />
⎣⎣ ⎢⎢ 1 ⎦⎦ ⎥⎥<br />
€<br />
; image:<br />
⎡⎡ 1⎤⎤<br />
⎢⎢ ⎥⎥<br />
v 1 ⎢⎢<br />
0<br />
⎥⎥<br />
⎣⎣ ⎢⎢ 0⎦⎦<br />
⎥⎥<br />
+ (v 2 + 2 v 3 )<br />
⎡⎡ 0⎤⎤<br />
⎢⎢ ⎥⎥<br />
⎢⎢<br />
3<br />
⎥⎥<br />
⎣⎣ ⎢⎢ 3⎦⎦<br />
⎥⎥<br />
;<br />
nullity( T ) = 1 ; rank( T ) = 2<br />
8.<br />
€<br />
€<br />
⎡⎡<br />
x(t) = 4 t + 1 ⎤⎤<br />
⎢⎢ ⎥⎥ e −t<br />
⎣⎣ − 4 t ⎦⎦<br />
€
9. a)<br />
P =<br />
⎡⎡ 1 5 ⎤⎤<br />
⎢⎢ ⎥⎥ b)<br />
⎣⎣ −1 −1 ⎦⎦<br />
⎡⎡ 1 ⎤⎤<br />
x(t) = c 1 ⎢⎢ ⎥⎥ e −t ⎡⎡ 5 ⎤⎤<br />
+ c<br />
⎣⎣ − 1<br />
2 ⎢⎢ ⎥⎥ e 3t<br />
⎦⎦ ⎣⎣ − 1⎦⎦<br />
10. X ( t ) = − ½ sin 2t + 2 sin t<br />
€<br />
€<br />
11.<br />
y = c 1 e t + c 2 e 4t − 2 3 t e t + 3 2<br />
€<br />
12. the determinant is zero
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