MATH 251Spring 2003Final ExamMay 8, 2003NAME :ID :INSTRUCTOR :There are 15 questions on 5 pages. Please read each problem carefully before startingto solve it. For each multiple choice problem 4 answers are given, only one of which iscorrect. Mark only one choice. For partial credit questions, all work must be shown -credit will not be given for an answer unsupported by work.NO CALCULATORS ARE ALLOWED.PLEASE DO NOT WRITE IN THE BOX BELOW.1:2:3:4:5:6:7:8:9:10:11:12:13:14:15:Total:MATH 251 Spring 2003 -Final Exam-1. (6 points) Consider the undamped system of a 7kg mass hanging from a spring. Anexternal force of 3 sin(10t) newtons is applied to the system, which then enters reso-nance. What is the spring constant?(a) 7(b) 10(c) 100(d) 7002. (6 points) What is the inverse Laplace transform of F (s) = e−3s1s+2?(a) u3(t) +12δ(t − 3)(b) u3(t)e2t+3(c) u3(t)e−2t+6(d) δ(t − 3)e−2t3. (6 points) What is the partial fraction expansion of7s−2s2(s−2)?(a)−3s+1s2+1s−2(b)1s+3s2+1s−2(c)−3s+1s2+3s−2(d)1s2+1s−24. (6 points) Consider the following two differential equations:I y00+ ay0+ by = 0 y(0) = 0 y0(0) = 2II y00+ ay0+ by = 0 y(0) = 0 y(π) = 2Where a, b ∈ R Which of the following statements are true?(a) Both I and II always have a unique solution on some interval.(b) Only I always has a solution on some interval.(c) Only II always has a solution on some interval.(d) None of the above.5. (6 points) What is the stability of the equilibrium solution y(t) = 3 for the followingautonomous differential equation?y0= y(y − 3)(y + 3)(a) Stable.(b) Unstable.Page 2 of 5MATH 251 Spring 2003 -Final Exam-(c) Semi-stable.(d) None of the above.6. (6 points) Which of the following equations is exact?(a) teytdydt+ yeyt+ 2t = 0(b) (yeyt+ 2t)dydt= teyt(c) (yeyt+ 2t)dydt+ teyt= 0(d) teytdydt+ yeyt− 2t = 07. (6 points) Which of the following graphs would have a fourier series consisting onlyof sine terms? Note: only one period of each function is shown.a)c)b)d)8. (6 points) Which of the following pair of functions are not linearly independant?(a) y1(t) = t and y2(t) = 1.(b) y1(t) = sin 2t and y2(t) = cos 2t.(c) y1(t) = e3tand y2(t) = e3t−2.(d) y1(t) = e−2tand y2(t) = te−2t.9. (16 points)Page 3 of 5MATH 251 Spring 2003 -Final Exam-(a) What is the interval on which a solution the following differential equation iscertain to exist?t2y00+ ty0− y = 0 y(1) = 2 y0(1) = 0(b) Given y(t) = t is a solution to the above differential equation (you need not checkthis) what is the general solution?10. (14 points) Consider the following nonlinear system.x0= xy − 3xy0= 2xy + 2y(a) Find the linearized matrix for this system.(b) Find the critical points.(c) Chose one of the critical points and state what is its type and stability.11. (14 points) Consider the following function, defined on the interval (0,2).f(x) =(x 0 < x < 12 1 6 x < 2(a) Graph the even, period 4, extension of f (x) on (−4, 4).(b) Graph the odd, period 4, extension of f (x) on (−4, 4).(c) Which of the above two has a cosine series, and which has a sine series?(d) What does the fourier series representing part b (the odd extension) converge toat x = −2, x =12and x = 3?12. (12 points) Consider the following partial differential equationuyy+ 2uy= y2uxx(a) Separate this equation into two ordinary differential equations.(b) Translate the following boundary conditions on the above partial differentialequation to conditions on the ordinary differential equations found above.u(0, y) = 0 u(L, y) = 013. (14 points) For the following boundary problem find all positive eigenvalues andtheir corresponding eigenfunctions.X00+ λX = 0 X(0) = 0 X0(π) = 0Page 4 of 5MATH 251 Spring 2003 -Final Exam-14. (14 points) Consider a thin rod of length 10cm with thermal diffusivity α2= 2cm2sandperfectly insulated ends. Using the variable x as the distance from the left end of therod the initial temperature of this rod isf(x) = 3 + cosπx5− 5 cosπ2(a) Construct the partial differential equation for this situation, also give boundaryconditions.(b) Solve this partial differential equation.(c) Determine the steady-state temperature.15. (18 points)(a) Find the general solution to the following system.X0(t) =0 1−1 −2X(b) Find the specific solution which meets the following initial condition.X(0) =21Page 5 of
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