MATH 251Examination I INovember 2, 2009Name:Student Number:Section:This exam has 11 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 answers notsupported by work.You may not use a calculator on this exam. Please turn off and put away yourcell phone.The last sh eet of the booklet contains a table of Laplace transforms and may be detached.Do not write in this boxQuestion Score1–67891011TotalMATH 251 Examination II November 2, 20091. (5 points) Match the phase portraits with their names by entering the numbers in the blanksbelow.1 2 34 5 6(a) node(b) saddle(c) spiral(d) center(e) proper node2. (5 points) Consider the mechanical system represented by the ODE:y′′+ 4y′+ 5y = 0.The system is(a) und erdamped(b) critically damped(c) overdamped(d) in resonancePage 2 of 9MATH 251 Examination II November 2, 20093. (5 points) Let f(t) = u(t − 2)et−2sin(3t − 6), whereu(t − c) = uc(t) =0 if t < c1 if c ≤ tThe Laplace transform of f(t) is(a) e−2s1s−13s2+9(b) e−2s3(s−1)2+9(c) e−2s1s−1ss2+9(d) e2ss−1(s−1)2+94. (5 points) Given a matrix A =4 −60 −2. Then the origin00of x′= Ax is(a) asymptotically stable sp iral(b) unstable saddle point(c) stable center(d) unstable spiral pointPage 3 of 9MATH 251 Examination II November 2, 20095. (5 points) EvaluateZ∞0e−stt dt, (s > 0)(a) −s(b) s−1(c) e−s(d) s−26. (5 points) Match the five names f or the critical point 0 of a homogeneous linear system ofODE’s x′= Ax with the five general solutions given in the table below by placing one of theletters N, S, P, C, or L in each of the five blanks. (Each letter should be used exactly once.)(i) c1et1−1+ c2e−2t11N Node(ii) c1etcos t2 sin t+ c2etsin t2 cos tS Saddle(iii) c1e−t12+ c2e−2t−21P Proper node(iv) c1e2t10+ c2e2t01C Center(v) c1cos t2 sin t+ c2sin t2 cos tL spiraLPage 4 of 9MATH 251 Examination II November 2, 20097. Consider the spring-mass system with damping and external force that is modeled by thefollowing ODE:y′′+ γy′+ 4y = F0cos(ωt) .(a) (8 points) If γ = 0 and F0= 0, then what is the amplitude of the disp lacement y of thesystem with y(0) = 1, y′(0) = −2 ?(b) (4 points) If F0= 0, then what is the smallest value of γ for which the object crosses itsequilibrium position a fin ite number of times.(c) (4 points) If F0= 3, then find a value for γ and a value for ω so th at th e system is inresonance.Page 5 of 9MATH 251 Examination II November 2, 20098. In the following parts determine the form of a p articular solution with as few constants aspossible. Do NOTsolve for the undetermined coefficients!(a) (4 points) y′′+ y′= t2+ e−t(b) (4 points) y′′+ 2y′+ y = te−t(c) (4 points) y′′+ 2y′+ y = t2e2tcos(4t)Page 6 of 9MATH 251 Examination II November 2, 20099. (a) (6 points) Express the following function by means of a single formula using unit-stepfunctions u(t − c), with various choices of c:f(t) =t for 0 ≤ t < 2,2 for 2 ≤ t < 4,0 for 4 ≤ t(b) (6 points) Compute the Laplace transform ofg(t) = t2+ u(t − 3)et− u(t −π2) cos(t)Page 7 of 9MATH 251 Examination II November 2, 200910. (a) (5 points) Assume that the acceleration due to gravity is g = 10 meters/sec2.An object whose mass is 2 kg stretches a spring 4 meters to equilibrium.At time t = 0 the object is released 1 meter above its equilibrium position with a downwardvelocity of 2 meters/sec. At time t = 3 a constant force of 5 Newtons in the upwarddirection is applied. Write an ODE and initial conditions that represent this spring-masssystem. DO NOT SOLVE IT.(b) (10 points) Solve using Laplace transform:y′′+ 2y′+ 2y = δ(t − 5), y(0) = 1, y′(0) = 2,where δ(t) is the Dirac delta function.Page 8 of 9MATH 251 Examination II November 2, 200911. (15 points) Solve the following linear homogeneous system of first order ODE’s:x′=2 −15 −4x, x(0) =40Page 9 of
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