18.034 FINAL EXAM MAY 20, 2004 Name: Problem 1: /10 Problem 2: /20 Problem 3: /25 Problem 4: /15 Problem 5: /20 Problem 6: /25 Problem 7: /10 Problem 8: /35 Problem 9: /40 Problem 10: /10 Extra credit Total: /200 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 3 hours 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. 112345678910112 18.034 FINAL EXAM Table of Laplace Transforms y(t) Y (s) = L[y(t)] . y(n)(t) snY (s) − (y(n−1)(0) + ··· + sn−1y(0)) . tn n!/sn+1 . tny(t) (−1)nY(n)(s) . cos(ωt) s/(s2 + ω2) . sin(ωt) ω/(s2 + ω2) . eaty(t) Y (s − a) . y(at), a > 0 1 a Y (s/a) . S(t − t0)y(t − t0), t0 ≥ 0 e−st0Y (s) . δ(t − t0), t0 ≥ 0 e−st0 . (S(t)y) ∗ (S(t)z) Y (s)Z(s) . y(t), y(t + T ) = y(t) 1 1−e−sT � T 0 e−sty(t)dt� � � � � 18.034 FINAL EXAM 3 Name: Problem 1: /10 Problem 1(10 points) Two objects of mass m are connected to a rigid base and to each other as shown on the previous page. The spring connecting each object to the base has constant k, and the spring connecting the objects to each other has constant 2k. Denote by x1 the displacement of the object on the left from equilibrium (displacement to the right = positive displacement). Denote by x2 the displacement of the object on the right from equilibrium (displacement to the right = positive displacement). Denote ω = k/m. (a)(5 points) Find a system of 2nd order linear ODEs satisfied by x1 and x2 of the form, x��x11 x��= Ax2 . 2 In other words, find the matrix A.4 18.034 FINAL EXAM Name: Problem 1, contd. (b)(5 points) Introduce new variables v1 = x�and v2 = x2�. Find a system of 1st order linear ODEs 1 satisfied by x1, v1, x2 and v2 of the form, ⎡⎤ ⎡ ⎤ x�1 x1 ⎢⎢⎣ v�1 x�2 ⎥⎥⎦ = B ⎢⎢⎣ v1 x2 ⎥⎥⎦ . v�2 v2 In other words, find the matrix B. Extra credit(2 points) What is the relationship of pA(λ) and pB(λ)?18.034 FINAL EXAM 5 Name: Problem 2: /20 Problem 2(20 points) Consider the ODE, y(t)� +2 y(t) = 3e−t3/3, t > 0 t(a)(5 points) Find an integrating factor. (b)(10 points) Find the general solution. (c)(5 points) Find the unique solution that has an extension to a continuous function on [0, ∞).6 18.034 FINAL EXAM Name: Problem 3: /25 Problem 3(25 points) A basic solution pair of the homogeneous linear 2nd order ODE, 2t 1 y�� + t2 − 4y� − 16(t2 − 4)2 y = 0, t > 2 is given by {y1(t), y2(t)}, t − 2 t + 2 y1(t) = t + 2, y2(t) = t − 2. (a)(10 points) Compute the Wronskian W [y1, y2](t).7 18.034 FINAL EXAM Name: Problem 3, contd. (b)(15 points) Use variation of parameters to find a particular solution of the inhomogeneous ODE, 2t 1 y�� + t2 − 4y� − 16(t2 − 4)2 y = 1.8 18.034 FINAL EXAM Name: Problem 4: /15 Problem 4(15 points) Using the method of undetermined coefficients and the exponential shift rule, find a particular solution of the inhomogeneous linear 2nd order ODE, y�� + 5y� + 6y = −4te−3t .�18.034 FINAL EXAM 9 Name: Problem 5: /20 Problem 5(20 points) On the interval [0, 2), let f(t) = t + 1. Denote by f�(t) the even extension of f(t) as a periodic function of period 4. Denote by FCS[f�] the Fourier cosine series of f�(t). (a)(5 points) Graph FCS[f] on the interval [−4, 4]. Make special note of all discontinuities and the actual value of FCS[ f�] at these points.��� 10 18.034 FINAL EXAM Name: Problem 5, contd. (b)(10 points) An orthonormal basis for the even periodic functions of period 4 is, 1 1 φ0(t) =2, φn(t) = √2 cos(nπt/2), n = 1, 2, 3, . . . Compute the Fourier coefficients, � 2 an = � �f, φn� = f(t)φn(t)dt, −2 and e xpress your answer as a Fourier cosine series, f(t) = a0 + ∞an cos(nπt/2).2 √2 n=1 Don’t forget to compute a0. Extra credit(3 points) Plug in t = 0 to get a formula for the series, � 1 (2m + 1)2 . ∞m=0� � 18.034 FINAL EXAM 11 Name: Problem 6: /25 Problem 6(25 points) Let f(t) be the piecewise continuous function, 0, 0 < t < 1 f(t) = e−3(t−1), t ≥ 1 Let y(t) be the continuously differentiable and piecewise twice-differentiable solution of the following IVP, ⎧ ⎨ y�� + 5y� + 6y = f(t), y(0) = 0,⎩ y�(0) = 0 Denote by Y (s) the Laplace transform, L[y(t)] = ∞ e−st y(t)dt. 0 (a)(5 points) Compute the Laplace transform of the IVP and use this to find an equation that Y (s) satisfies. (b)(10 points) Solve the equation fo Y (s) and find the partial fraction decomposition of your answer. Use the Heaviside cover-up method to simplify the partial fraction decomposition.12 18.034 FINAL EXAM Name: Problem 6, contd. (c)(10 points) Find y(t) by computing the inverse Laplace transform. Question:(Not to be answered) Is there a simpler solution than using the Laplace transform? If so, you can use this to double-check your answer.18.034 FINAL EXAM 13 Name: Problem 7: /10 Problem 7 Let A be the real 3× 3 matrix, ⎡ ⎤ 2 1 0 A = ⎣ 0 −1 1 .⎦ 0 0 2 (a)(3 points) Compute the characteristic polynomial pA(λ) = det(λI − A). (b)(7 points) For each eigenvalue, find an eigenvector (not a generalizaed eigenvector ).14 18.034 FINAL EXAM Name: Problem 7, contd. Extra credit(3 points) For one of the eigenvalues, the eigenspace is deficient. Find a generalized eigenvector that is not an eigenvector.18.034 FINAL EXAM 15 Name: Problem 8: /35 Problem 8(35 points) The
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