G251ex2(fa05)

# G251ex2(fa05) - Math 251 Sections 1/2 November 7, 2005...

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Math 251 – Sections 1/2 November 7, 2005 Second Exam ANSWER KEY There are 9 questions on this exam. Question 1 is worth 20 points. Questions 2 through 9 are worth 10 points each. The total number of points is 100. If a question has multiple parts, then the points assigned to the question are divided equally among the parts, unless otherwise indicated. Where appropriate, show your work to receive credit; partial credit may be given. The use of calculators, books, or notes is not permitted on this exam. Please turn oﬀ your cell phone. Time limit 1 hour and 15 minutes.

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1. i. One of the following ODE’s represents the displacement y ( t ) in a spring-mass system with resonance? Circle it. 3 y 00 +108 y = 4 sin 6 t 3 y 00 +108 = 2 sin 6 t 3 y 00 +108 y = 6 cos 3 t 3 y 00 +111 y 0 +108 y = cos 6 t ANS. Frequency of external force and natural frequency match in the last equation. So its the ﬁrst one. ii For a spring-mass system system with mass equal to 4 kg, spring contant equal to 9 N/m, which damping constant γ causes critical damping? ANS. γ = 4 mk = 6 iii For a spring-mass system system with mass equal to 4 kg, spring contant equal to 9 N/m, damping constant equal to 5, and external force equal to sin t what does the transient part of the solution look like? (Do not determine any constants.) ANS. Since γ 2 - 4 mk is negative the transient part of the solution is a combination of sine and cosine multiplied by a decaying exponential. iv. What is the deﬁnition of the Laplace transform of g ( t )? ANS. F ( s ) = Z 0 e - st f ( t ) dt v What is the following Laplace transform: L{ t 2 e 2 t } ANS. 2 ( s - 2) 3 vi. Circle all the functions among the following that have a Laplace transform : e t 3 t 1 - t | t - 1 | t - 1 e t 1 / 2 ANS. The last two. vii. Suppose that the Laplace transform of y is Y . If y (0) = 2 and y 0 (0) = - 3, then ﬁnd the Laplace transform of y 00 . ANS. s ( sY - 2) + 3 viii. Suppose that a force of - 3 δ ( t - 1) acts on an object of mass 1 kg. If y (0) = 2 and y 0 (0) = 2, then ﬁnd y 0 (2).
ANS. y 0 (2) = - 1. ix Suppose that the homogeneous linear system x 0 = A x has only one eigenvalue r 1 = 3 and that all eigenvectors are multiples of a single vector. Find the solution satifying the IVP: x (0) = ± - 2 1 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 Penn State.

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G251ex2(fa05) - Math 251 Sections 1/2 November 7, 2005...

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