MATH 251Examination I IApril 7, 2010FORM AName:Student Number:Section:This exam has 10 questions for a total of 100 points. In order to obtain full credit forpartial credit problems, all work must be shown. For other problems, points might bededucted, at the sole discretion of the instructor, for an answer not supported by areasonable amount of work. T he point value for each question is in parentheses to the right ofthe question number. A table of Laplace transforms is attached as the last page of the exam.Please turn off and put away your cell phone.You may not use a calculator on this exam.1thru4:5:6:7:8:9:10:Total:Do not write in this box.MATH 251 EXAMINATION II April 7, 20101. (5 points) A mass-spring system is initially resting at its equilibrium position w ith no forcesacting on it. Between times t = 2 and t = 5 seconds a constant external downward force of 10Newtons is applied. At t = 6 seconds the mass is struck with a hammer in such a fashion thatan upward momentum of magnitude 20 kg-m/s is introdu ced to the system at that time. Giventhat the downward direction is positive, which of the following F (t) represents the combinationof these two forces?(a) F (t) = 10u2(t) − 10u5(t) − 20δ(t − 6)(b) F (t) = 10u2(t) + 10u5(t) − 20u6(t)(c) F (t) = 10u5(t) − 10u2(t) − 20(1 − u6(t))(d) F (t) = 10δ(t − 2) − 10δ(t − 5) − 20u6(t)2. (5 points) Which system of first order linear equations below is equivalent to the third orderlinear equationy′′′+ y′′− 2y′+ 3y = 0?(a)x′1= x2x′2= x3x′3= −x1+ 2x2− 3x3(b)x′1= x1x′2= x2x′3= 3x1− 2x2+ x3(c)x′1= x1x′2= x2x′3= x1− 2x2+ 3x3(d)x′1= x2x′2= x3x′3= −3x1+ 2x2− x3Page 2 of 9MATH 251 EXAMINATION II April 7, 20103. (5 points) Suppose a certain linear system x′= Ax, w here A is a matrix of real numberswith nonzero eigenvalues, has at least onenonzero solution that does not reach a limit both ast → ∞, and as t → −∞. Then the critical point (0, 0) must be(a) un s table.(b) asymptotically stable.(c) (neutrally) stable, but not asymptotically stable.(d) having indeterminate stability.4. (5 points) All of the following systems of linear equations have exactly one critical point, at(0, 0). In three of the systems, the stability of (0, 0) are identical. In which system does (0, 0)have a different stability classification?(a) x′=1 00 −2x(b) x′=0 −4−1 0x(c) x′=0 1−9 0x(d) x′=5 00 5xPage 3 of 9MATH 251 EXAMINATION II April 7, 20105. (14 points) Find the inverse Laplace transform of each function given below .(a) (7 points) F (s) =√7 +4s + 10s2− 4s + 20(b) (7 points) F (s) = e−s1s3+ s2Page 4 of 9MATH 251 EXAMINATION II April 7, 20106. (12 points) Rewrite the follow ing piecewise continuous function f(t) in terms of th e unit-stepfunction. Then find its Laplace transform.f (t) =6eπt, 0 ≤ t < 42t2− 3t, 4 ≤ t.Page 5 of 9MATH 251 EXAMINATION II April 7, 20107. (16 points) Use the Laplace transform to solve the following initial value problem.y′′+ 16y = u7(t) − δ(t − 13), y(0) = 0, y′(0) = 1.No credit will be given if the Laplace transform is not used to solve this problem.Page 6 of 9MATH 251 EXAMINATION II April 7, 20108. (14 points)(a) (9 points) Solve the initial value problemx′=1 32 2x, x(0) =04.(b) (3 points) Given that x(0) =9β, and limt→∞x(t) =00, find the value of β.(c) (2 points) Classify the type and s tability of the critical point at (0, 0).Page 7 of 9MATH 251 EXAMINATION II April 7, 20109. (12 points) In the parts below, consider a certain system of two first order linear d ifferentialequations in two unknowns, x′= Ax, where A is a matrix of real numbers.(a) (4 points) Suppose one of the eigenvalues of the coefficient matrix A is r = 7 + 3i, whichhas a corresponding eigenvector−2 − 4i1. Write down the system’s real-valuedgeneralsolution.(b) (2 points) State the type and stability of the critical point (0, 0) of the system in (a).(c) (4 points) Consider a different sys tem. Suppose its coefficient matrix A is s uch that it hasthe following matrix-vector p roducts A10=−20and A−1−1=22. Writedown the system’s general solution.(d) (2 points) State the type and stability of the critical point (0, 0) of the system in (c).Page 8 of 9MATH 251 EXAMINATION II April 7, 201010. (12 points) Consider the nonlinear system:x′= x2− xyy′= xy + 2y2− 6y(a) (4 points) The system has 3 critical points. One of the critical points is (0, 3). Find theother 2 critical points.(b) (8 points) Linearize the system about the point (0, 3). Classify the type and s tability ofthe critical point at (0, 3) by examining the linearized system. Be sure to clearly state thelinearized system’s matrix and its eigenvalues.Page 9 of
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