# Exam2Fall95 - 65.2400 TEST 2 November 6 1995 Name Section...

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Unformatted text preview: 65.2400 TEST 2 November 6, 1995 Name Section Please answer all questions, showing your work in detail. You may use one 8.5 by 11 sheet of notes while taking the examination. No other reference material, books, notes, or calculators are permitted. NAME _____________________-____-_____ SECTION __________________ _- 4. (a) (10 points) A certain spring-mass system satisﬁes the equation 9U” + 16u = 0. It is set in motion with initial conditions u(0) = 3, u’(0) = v. The amplitude of the resulting motion is observed to be four. What is u? (b) (10 points) Find the eigenvalues 0 and the eigenfunctions X of the boundary value problem X” +aX = 0, X’(O) = 0, X(8) = 0. You may assume that a > 0. In other words, ﬁnd the nonzero solutions of this problem. NAME ______________-__-_____-_______- SECTION ___________________ __ (C) (5 points) A certain spring-mass system satisﬁes the equation Zu” + ku = cos 3t. Find the value of k for which resonance occurs. (d) (5 points) Find the steady state part of the solution of the following heat conduction problem. 4uzz = ut u(0, t) = 20, uz(1o, t) = —3 u(:r,0) = 3m NAME _________--_-_______-____________ SECTION ___________________ __ 5. (20 points) Consider the heat conduction problem um=ut,0<\$<10,t>0 man=0tuaw=o¢>o w,0<x<5 “Lm‘{0,5<x<m The functions wk(x, t) = exp(—k27r2t / 100) sin(k7r:r / 10) are known to satisfy the partial differential equation and boundary conditions for k = 1, 2, 3, Find a function that ALSO satisﬁes the initial condition. NOTE. You should evaluate the coefﬁcients, but do not try to simplify sin(k7r/2) or similar expressions. Write out the ﬁrst three nonzero terms in your solution in detail. LAB TEST 2 65.2400 — 13, 14, 15 November 2, 1995 NAME .___ _____ .____..____._._ SECTION ._________.__.__ 1. Consider a spring-mass system that satisﬁes the initial value problem u” + (1/4)ul + 21/. = 0, 11(0) = 1,u'(0) = 3 (a) Solve this problem and write the solution below. (b) Plot the graph of the solution for 0 S t S 20 on the screen and draw a reasonable sketch of the graph here. Put a scale on the u axis. U. Jr 10 IS Io (c) Find (to five decimal places) the smallest value T such that {u(t)| < 0.01 for t > T. T: (d) Let ul, uz, and U3 be the u—coordinates of the first three local maximum points to the right of t = 0. Also let r1 = 112/111 and r2 = u3/uz. Determine 7‘1 and r2, correct to at least two decimal places. T1=________. T2: NOTE: You may do this either by making estimates from a graph, or by performing appropriate calculations. In either case, we need at least two decimal places correct. ‘2. Let f(I) = I2(4—;r) for O s r g 4. Consider the Fourier sine series for this function with period eight. (a) Evaluate the Fourier coefficients bk. Leave your answers in terms of sin k7r and cos k7r if you wish. bk _.__._.________ (b) Find the numerical values of b4 and b7. b4 — ______.___..______ b7 = ______..__._ (c) Let ‘sn(1:) be the n‘h partial sum in the Fourier series, and let err(h, :r) = [531(1) - f(r)|. Plot 87‘7‘(10,I) for 0 g 2: S 4. Sketch the plot below. Put a scale on the vertical axis. t en- (l0,x\' I (d) Estimate from the plot the maximum value of err(10,:1:) for 0 S 1: S 4 and the value of I where this maximum occurs. I = _______ max can-(10, z) = (e) Find the smallest n for which err(n, I) < 0.01 for all x in 0 S 1' g 4. ...
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Exam2Fall95 - 65.2400 TEST 2 November 6 1995 Name Section...

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