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Unformatted text preview: Math 251 Spring 2004 Final Exam ANSW ERS 1. a) True; b) True; c) False; d) True; e) False; f) True
2. a) µ(t) = t−3 ; b) c = 3; c) Y (t) = Ate−2t + Bt2 + Ct + D.
3. a) T = √
2π
(sec); b) µ = 5 (rad/sec); c) It is critically damped.
3 4. y(t) = u1 (t)(t − 1)e3(t−1) − 2u3 (t)(t − 3)e3(t−3)
5. x′1 = x2
x′2 = −5 x1 + x2
2 or 1
3
e4t + C2
2
1
b) It is an unstable saddle point.
c) α = −1
d) 6. a) x(t) = C1 0 1
1 x′ = −5
2 x e−t cos t − 4 sin t
e3t
4 cos t + sin t
b) It is an unstable spiral point. 7. a) x(t) = 8. a) The critical points are (0, −2), (1, 1), (−2, −2).
b) (0, −2) is an unstable saddle point, (0, 0) is an unstable spiral point, and (−2, −2) is an
asymptotically stable node.
9. λX ′′ − X ′ + 5x3 X = 0
λT ′ − T = 0 n2
nx
1 4 9
, n=
10. The eigenvalues are λ = . , , ..., , ... Their eigenfunctions are yn (x) = Cn cos
4 4 4
4
2
1, 2, 3, 4, ...
b) Yes, λ = 0 is an eigenvalue. Any nonzero constant function is a corresponding eigenfunction.
1 −1
1
mπx
mπx
1
(−2 − x) cos
dx +
dx +
x cos
2 −2
2
2
−1
Just state that am = 0 for all m (by noting that f (x) is an odd function). 2 11. For m = 0, 1, 2, 3, ..., am =
or 1 (2 − x) cos 1
2
nπx
nπx
nπx
dx +
dx +
dx
x sin
(2 − x) sin
2
2
2
−2
−1
1
2
1
nπx
nπx
dx +
dx , (taking advantage of the fact that the
(2 − x) sin
or
bn =
x sin
2
2
1
0
product f (x) sin nπx is an even function).
2 For n = 1, 2, 3, ..., bn = 12. u(x, t) = 2e −5π 2 t
16 sin 1
2 −1 (−2 − x) sin πx
2
− e−5π t sin πx
4 2 mπx
dx
2 ...
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This note was uploaded on 07/23/2008 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.
 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Partial Differential Equations

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