s251final(su03) - Answers to Summer 2003 Final exam. Q#:1 A...

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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.

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s251final(su03) - Answers to Summer 2003 Final exam. Q#:1 A...

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