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Unformatted text preview: Answers to Summer 2003 Final exam.
Q#:1
A A D.E. that satisﬁes these requirements is y = (y − 1)(y − 2)(y − 3).
B A D.E. that satisﬁes these requirements is y + 4y + 4y = 3 cos t − 4 sin t.
C A D.E. that satisﬁes these requirements is 2xy + ey + (x2 + xey ) dx = 0.
dy
t
Q#:2 The answer, for both methods, is y(t) = 2 et − 1 et + 5 e−t
4
4
Q#:3 A There is an equilibrium solution at v = 15, which is stable, and another at v = −15, which is unstable.
1
B The solution is t + c = 120 [ln(30 + 2v) − ln(30 − 2v)].
Q#:4
A
B
C
D The D.E. is y + 7y + 10y = cos t − u3π (t) cos t, with initial conditions y(0) = 0 and y (0) = 2.
9
7
8
9
7
2
5
y(t) = 130 cos t + 130 sin t + 15 e−2t − 47 e−5t + u3π (t) − 130 cos t − 130 sin t − 15 e−2(t−3π) + 78 e−5(t−3π) .
78
9
8 −2π
57 −5π
8 −8π
57 −20π
2 −2π
5 −5π
− 78 e
and y(4π) = 15 e
− 78 e
− 15 e
+ 78 e
y(π) = − 130 + 15 e
√
The natural frequency is ωo = 10 rad
s Q#:5
5
1
+ e−3t
.
−3
−1
B This critical point, the only one, is a stable node. It is assymptotically stable.
A The solution is X(t) = −e−t Q#:6
A The critical points are P = (1, −1), Q − (−1, −1), R = (2, −2) and S = (2, 2).
2x
2y
B The linearized matrix is A(x, y) =
.
y+1 x−2
C
2 −2
X. It is a saddle point, which is unstable.
• Point P: X =
0 −1
−2 −2
X. It is a stable node, which is assymptotically stable.
• Point Q: X =
0 −3
4 −4
X. It is an unstable node.
• Point R: X =
−1 0
4 4
X. It is a saddle point, which is unstable.
• Point S: X =
3 0
Q#:7
2 A The eigenvalues are λn = (2n−1) , where n ∈ N.
4
B The eigenfunctions are fn = cos( (2n−1)t ), where n ∈ N.
2
Q#:8 A B C The coeﬃcients for the odd series are given by an =
˜
˜
˜
D f (−3) = 0, f (0) = 0 and f (2) = −1.
Q#:9 1
3 3
−3 f (x) sin( nπx ) dx.
3 A The two ordinary D.E.s are t2 T (t) − λT (t) = 0 and xX (x) − λX(x) = 0.
B The boundary conditions become X(0) = X(1) = 1.
Q#:10
u(x, t) = 20 + 10x + 5e−10π t sin πx − 10 exp
2 −45π 2 t
2 sin 3πx
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
 Differential Equations, Equations, Partial Differential Equations

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