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Unformatted text preview: Math 251 November 7, 2005 Ans Key to 2nd Exam Name There are 8 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question or at the beginning of each part where there are multiple part 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. Question Score 1 26pt 2 12pt 3 10pt 4 10pt 5 10pt 6 10pt 7 12pt 8 10pt Total 100pt 1. i. 2pt One of the following ODE’s represents the displacement y ( t ) in a springmass system with resonance? Circle it. y 00 + 36 y = 4 sin 6 t y 00 + 36 = 2 sin 6 t y 00 + 10 y + 34 y = 6 cos 3 t y 00 + 37 y + 36 y = cos 6 t ANS. Resonance occurs when frequencey of external force equals natural frequency. This happens with the FIRST ODE ii 4 pt For a springmass system system with mass equal to 4 kg, spring contant equal to 9 N/m, which damping constant γ causes critical damping? ANS. γ critical = √ 4 mk = p 4(4)(9) = 12 iii. 4 pt For a piecewise continuous function g ( t ) that has exponential growth, what is the definition of the Laplace transform of g ( t )? ANS. L{ g(t) } = Z ∞ e st g(t) dt iv 4 pt What is the following Laplace transform: L{ te 2 t } ANS. There are two procedures to the correct answer. Either, since L{ t } = 1 s 2 , L{ e 2t t } = 1 (s 2) 2 . Or, since L{ e 2t } = 1 s 2 , L{ te 2t } = 1 (s 2) 2 . Give 2pt for writing correct starting point. 2pt for correct answer. Do not give more than 2pt for correct answer without correct starting point! v. 4 pt Suppose that the Laplace transform of y is Y . If y (0) = 2 and y (0) = 3, then find the Laplace transform of y 00 . ANS. Give 1pt for L{ y } = sY 2 Give 3pt for L{ y 00 } = s(sY 2) ( 3) = s 2 Y 2s + 3 vi. 4 pt Suppose that a force of 3 δ ( t 1) acts on an object of mass 1 kg. If y (0) = 2 and y (0) = 2, then find y (2). ANS. This force decrease momentum (equal to velocity, here) by 3 units at t = 1. It does nothing at any other time. So y (2) = 2 3 = 1 (There is no way to earm partial credit here). vii 4 pt Suppose that the homogeneous linear system x = A x has only one eigenvalue r 1 = 3 and that all nonzero vectors eigenvectors. Find the solution satifying the IVP: x (0) = 2 1 . ANS. Give 2pt for indicating the solution is a multiple of 2 1 . Give another 2pt for if the multiple is correct: ie, x = e 3 t 2 1 . 1 2. In parts a c determine the form of a particular solution y p having the least number of unknown constants. DO NOT DETERMINE the unknown constants appearing in your answers in parts a and b ....
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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.
 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Partial Differential Equations

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