Page 1 of 7Student Name:Student Net ID:MATH 286 SECTION X1 – Introduction to Differential Equations PlusMIDTERM EXAMINATION 2October 23, 2013INSTRUCTOR: M. BRANNANINSTRUCTIONS• This exam is 60 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: 12 16 10 12 50Score:1. Consider the differential equationLy = y(4)+ 3y00− 4y = 0.(a) (3 points) Let y1(x), y2(x), y3(x), y4(x) be four solutions to the above ODE. Explainhow the Wronskian of these functions, W (x), can be used to determine if y1, y2, y3, y4are linearly dependent or linearly independent on R.(b) (9 points) Find four linearly independent solutions to the above ODE and provethat they are linearly independent by computing their Wronskian. Extra space isprovided for your solution on the following page.(Hint: It may be helpful to pick a specific value of x at which to evaluate W .)Student Net ID: MATH 286 G1 Page 3 of 7(Extra Space For Problem 1(b))Student Net ID: MATH 286 G1 Page 4 of 72. Consider a 7th order linear ODE Ly = F (x) with associated characteristic polynomialP (r) = r3(r − 1)2(r2− 2r + 2).(a) (6 points) If F (x) = 0, find the general solution to this equation.(b) (10 points) Use the method of undetermined coefficients to find a particular solu-tion to the ODE Ly = F (x) for the following choices of forcing term F (x). Youare not required to solve for the undetermined coefficients!1. F (x) = xe2x.2. F (x) = xex.3. F (x) = e2xsin x +x44.4. F (x) = x2exsin x.Student Net ID: MATH 286 G1 Page 5 of 73. Consider the following eigenvalue problem:y00+ 20y0− λy = 0; y(0) = y0(2) = 0.(a) (4 points) Show that if λ ≥ −100, then λ is not an eigenvalue.(b) (6 points) Show that if λ < −100, then λ is an eigenvalue if and only if λ =−100 − α2, where α > 0 is a solution to the equation α = 10 tan(2α).Student Net ID: MATH 286 G1 Page 6 of 74. Consider a simple RLC circuit with external voltage source E(t) (Volts), inductance L(Henries), capacitance C (Farads) and resistance R (Ohms). Recall that if I(t) denotesthe current (in Amps) passing through the circuit at time t, then I(t) satisfies the ODELI00+ RI0+1CI = E0(t).Assume for the remainder of the problem that R = C = L = 1 and E(t) = sin(ωt)(where ω 6= 0).(a) (5 points) Use the method of undetermined coefficients to derive an expressionfor the steady-state periodic current Isp(t) in the circuit.(b) (4 points) Show that the amplitude of Isp(t) is given byC(ω) =1p1 + (ω−1− ω)2.(c) (3 points) What frequency ω maximizes the amplitude of Isp(t)?Student Net ID: MATH 286 G1 Page 7 of 7(Extra work
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