MATH 251Examination I IJuly 30, 2007Name:Student Number:Section:This exam has 9 questions for a total of 100 points. In order to obtain full credit for partialcredit problems, all work must be shown. Credit will not be given for an answer notsupported by work. The point value for each question is in parentheses to the right of thequestion number. A table of Laplace transforms is attached as the last page of th e exam.Please turn off and put away your cell phone.You may not use a calculator on this exam.1:2:3:4:5:6:7:8:9:Total:Do not write in this box.MATH 251 EXAMINATION II July 30, 20071. (14 points) Consider a mass-spring system described by the equation3u′′+ 12u′+ ku = 0, k > 0.Answer th e followin g questions. Be sure to justify your answer. Full credit will not be givenwithout supporting work.(a) For what value(s) of k would the system be overdamped?(b) When k = 12, determine whether the system is over-, under-, or critically damped.(c) The system would oscillate when: k = 9 or k = 15.(Circle the correct value and justify your answer.)(d) Find the quasi-period of the system whose k-value you found in part (c).(e) True or False: When k = 3 the mass will never cross the system’s equilibrium positionmore than once.Page 2 of 9MATH 251 EXAMINATION II July 30, 20072. (5 points) A mass of 2 kg s tretches a spring 0.4 m. The system has no damping. At t = 0,the mass is pulled down 1 m from its equilibrium position and set in motion w ith an initialdownward velocity of 4 m/s. You may use g = 10 m/s2as the gravitational constant.(a) Find the Hooke’s constant, k, of the spring.(b) Set up, but do not solve, an initial value problem to find the system’s diplacementfunction u(t).3. (5 points) Given that L{f(t)} = F (s), what is L{e5ttf(t)}?(a)1s2(s − 5)F (s)(b)−1s − 5F′(s)(c)1s2F (s + 5)(d) −F′(s − 5)Page 3 of 9MATH 251 EXAMINATION II July 30, 20074. (14 points) Find the inverse Laplace transforms of(a)s2− 2s − 1s(s + 1)2(b) e−4s2s − 8s2+ 2s + 26Page 4 of 9MATH 251 EXAMINATION II July 30, 20075. (14 points) Rewrite the following piecewise continuous function f(t) in terms of the unit-stepfunctions. Then find its Laplace transform L{f (t)}.f(t) =e2t, 0 ≤ t < 48t − t2, 4 ≤ tPage 5 of 9MATH 251 EXAMINATION II July 30, 20076. (14 points) Use the Laplace transform to solve the initial value problemy′′+ 5y′− 6y = δ(t − 3), y(0) = 0, y′(0) = 2.No credit will be given if the Laplace transform is not used to solve this problem.Page 6 of 9MATH 251 EXAMINATION II July 30, 20077. (8 points) Consider the initial value problemy′′′− 2y′′+ y′+ 4y = e−2t+ δ(t − π), y(0) = 0, y′(0) = 1, y′′(0) = 0.Find L{y (t)} = Y (s), the Laplace transform of its solution. You do not need to simplify youranswer. Do notsolve for its inverse transform, y(t)!Page 7 of 9MATH 251 EXAMINATION II July 30, 20078. (a) (14 points) Find the general solution ofx′=2 64 4x.(b) Given the initial cond ition x(0) =6β, and suppose that limt→∞x(t) =00. Find thevalue(s) of β.Page 8 of 9MATH 251 EXAMINATION II July 30, 20079. (12 points) In each part below , consider a certain system of two first order linear differentialequations in two unknowns, x′= Ax.(a) Suppose that th e system’s general solution isx(t) = C121e2t+ C21−1e2t.Classify the type and stability of the system’s critical point at (0, 0).(b) Suppose one of the eigenvalues of the coefficient matrix A is −4 + 5i, which has a corre-sponding eigenvector1 + i−2i. Write down the system’s real-valuedgeneral solution.(c) Classify the type and stability of the critical point at (0, 0) for the system described in(b).(d) Suppose A has eigenvalues 2 and −3, classify the type and stability of the system’s criticalpoint at (0, 0).Page 9 of
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