18.034 EXAM 2 MARCH 17, 2004 Name: Problem 1: /30 Problem 2: /20 Problem 3: /25 Problem 4: /25 Total: /100 Instructions: Please write your name at the top of every page of the exam. The exam is closed book, closed notes, and calculators are not allowed. You will have approximately 50 minutes for this exam. The point value of each problem is written next to the problem – use your time wisely. Please show all work, unless instructed otherwise. Partial credit will be given only for work shown. You may use either pencil or ink. If you have a question, need extra paper, need to use the restroom, etc., raise your hand. Date : Spring 2004. 1Name: Problem 1: /30 Problem 1(30 points) A driven, damped harmonic oscillator satisfies the following ODE, y�� + 2by� + ω2 y = F cos(ωt), where b, ω and F are positive real numbers. (a)(10 points) Using the method of undetermined coefficients, find a solution of the form y(t) = C1 cos(ωt) + C2 sin(ωt). (b)(10 points) Let R be a positive real number, R ≤ F/(2bω) (this guarantees that yd(t) = R for some t > 0). There is a multi-valued function T = T (ω) for the set of positive numbers where thy(T ) = R. This can be made into a single-valued function Tn by specifying that Tn is the nsmallest positive number such that y(T ) = R . So T1 is the smallest positive number such that y(T1) = R, T2 is the smallest positive number greater than T1 such that y(T2) = R, e tc. For at least one choice of n > 0, find a formula for Tn(ω).(c)(10 points) Suppose that b, F and R are fixed, but ω is allowed to vary, F0 < ω ≤ .2bR Write T = Tn for some positive integer n. Let ω be a critical point of T (ω). Prove there is an equation, ωT (ω) = α, where α is a real number independent of b, F and R. Moreover, find an equation that α satisfies. (Remark You will see there are many choices for α. You are not responsible for “matching” choices of α to choices of n; just write down an equation that α satisfies). Extra credit(5 points) Prove the solutions of your equation for α give local minima of T (ω).Name: Problem 2: /20 Problem 2(20 points) In each case below, y1(t), y2(t) is a pair of solutions of a real, constant coefficient, linear homogeneous ODE in normal form. Determine the least degree of this ODE, and write down the ODE of this degree that the pair satisfies. (Hint: In each case, write down each nonzero solution as the real or imaginary part of eλtg(t) where g(t) is a polynomial. What does the degree of g(t), and the vanishing/nonvanishing of the imaginary part of λ tell you about the characteristic equation of the ODE? Remember, the ODE is a real ODE.) t(a)(5 points) y1(t) = 0, y2(t) = e . (b)(5 points) y1(t) = e−t , y2(t) = e−2t . t(c)(5 points) y1(t) = t, y2(t) = e . (d)(5 points) y1(t) = sin(2t), y2(t) = cos(3t).Name: Problem 3: /25 Problem 3(25 points) For a ce rtain linear ODE in normal form Ly a basic solution set of Ly = 0 is given by, 2t y1(t) = e , y2(t) = 2t2 + 2t + 1. (a)(10 points) Compute the Wronskian of this basic solution pair. Is your answer consistent with Abel’s theorem? (b)(15 points) Using the method of variation of parameters, find a particular solution of the inho-mogeneous ODE, 2tLy = t2 e .� � � �Name: Problem 4: /25 Problem 4(25 points) One solution of the linear ODE, y�� + a(t)y� + b(t)y = y�� + −2 − 2 ty + 4 ty = 0, 2tis the equation y1(t) = e . (a)(10 points) Le t y2(t) be a second solution. Define W (t) to be the Wronskian of y1(t), y2(t). This satisfies the differential equation W� = −a(t)W . Find a solution W (t) of this differential equation. (b)(15 points) For your solution W (t), solve the first-order linear ODE, y1(t)v� − y1(t)v = W (t). Plug in y2(t) = v to find a basic solution pair of the ODE, Ly =
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