Page 1 of 7Student Name:Student Net ID:MATH 286 SECTION G1 – Introduction to Differential Equations PlusMIDTERM EXAMINATION 2October 18, 2012INSTRUCTOR: M. BRANNANINSTRUCTIONS• This exam 50 minutes long. No personal aids or calculators are permitted.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page. There is a blank page at the endof the exam for rough work.• EXPLAIN YOUR WORK! Little or no points will be given for a correct answer withno explanation of how you got it. If you use a theorem to answer a question, indicatewhich theorem you are using, and explain why the hypotheses of the theorem are valid.• GOOD LUCK!PLEASE NOTE: “Proctors are unable to respond to queries about the interpretation ofexam questions. Do your best to answer exam questions as written.”Student Net ID: MATH 286 G1 Page 2 of 7Question: 1 2 3 4 TotalPoints: 15 15 12 12 54Score:1. Consider the ordinary differential equationLy = y(3)− 2y00+ 2y0= x + xex.(a) (5 points) Determine the complementary function yc(x) for the above ODE.(b) (8 points) Find a particular solution yp(x) to the above ODE.Student Net ID: MATH 286 G1 Page 3 of 7(c) (2 points) Write down the general solution to the above ODE.Student Net ID: MATH 286 G1 Page 4 of 72. Consider the ordinary differential equationLy = y00−1xy0+1x2y = x3.(a) (4 points) Verify that y1(x) = x and y2(x) = x ln |x| are solutions to the associatedhomogeneous ODELy = 0 (x 6= 0).(b) (5 points) Compute the Wronskian W (y1, y2) for the pair of functions y1, y2above.Are y1and y2linearly independent on the interval I = (0, ∞)? Why or why not?(c) (6 points) Find a particular solution toy00−1xy0+1x2y = x3(x > 0).(Hint: One possible approach is to use the variation of parameters method.)Student Net ID: MATH 286 G1 Page 5 of 73. (12 points) For the following endpoint problem, determine all eigenvalues λ ∈ R andtheir associated eigenfunctions:y00− 4y0+ λy = 0, y(0) = y(1) = 0.Student Net ID: MATH 286 G1 Page 6 of 74. Consider a mass-spring-dashpot system with mass m = 1kg, spring constant k = 4 N/mand dashpot damping constant β > 0 N s/m. Let x(t) denote the displacement (inmetres, at time t) of the mass from its equilibrium resting position.(a) (4 points) For what values of β is the system underdamped?For the remainder of the problem, assume that the dashpot is disconnected fromthe system (i.e., set β = 0) and that an external force F (t) = 2 sin ωt Newtons isapplied to the mass.(b) (2 points) At what forcing frequency ω will resonance occur in the forced system?(c) (6 points) Write down the general solution x(t) in this case.Student Net ID: MATH 286 G1 Page 7 of 7(Extra work
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