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Unformatted text preview: LABORATORY TEST ‘2 65.2400 — 07, 08, 09 March 28. 1994 NAME ________________..__________ SECTION _______________ I y 1. A certain springmass system corresponds to the differential equation 1 1
u/'+§u/+4u=0 (a) Solve this equation with the initial conditions u(0) = 2. u’(0) = 0. Plot the solution for O _<_ t S 30 and sketch the graph of u versus t on the axes belowi
u \o la 30 Now consider the related equation u” + éu’ + 421 = cos (St/2). (b) Solve this equation with the same initial conditions as before and plot the solution
for 0 g t g 40. Estimate from the graph the period and amplitude of the eventual steady oscillation.
Period __________ Amplitude _____ How do these compare with the period and amplitude of the forcing term, cos (5t/2)? (c) Compute u’ and plot u’ versus u parametrically. On the axes below sketch the part of the graph that corresponds to one cycle of the steady oscillation.
+ u’ N 65.2400  07, 08, 09 TEST 2 March 30, 1994 Name
Section 07 08 09
Please answer all questions, showing your work in detail.
You may refer to one sheet (8.5 by 11 inches) of notes. No other notes, books, references, calculators, etc. are permitted. Problems 3 and 4 count 25 points each; problem 5 counts 20 points. NAME ___ SECTION I 1 (a) Find the solution of the initial value problem 1/” + 42/ + 4y = 0, 9(0) = 2, y’(0) = —1 I (b) Consider the boundary value problem
X”+0X=O, X’(O)=0,X(5)=0 Find the eigenvalues and eigenfunctions, that is, find the values of a for which there
are nonzero solutions X (1:), and also ﬁnd the nonzero solutions. Assume that a > 0. NAME SECTION 2.Let
1, 0<x<1 f(z)={—1,1<x<2 I (a) Sketch the even extension of period 4 of f for —6 < a: < 6. (b) Find the Fourier series for the extended function in part (a). Determine a general
formula for the coefﬁcients, and write down the ﬁrst three nonzero terms explicitly. (c) To what value does the series converge when :1: = —1; when :1: = 2? NAME _______ SECTION f 3 (a) Use the Inethod of undetermined coefﬁcients to ﬁnd a particular solution of the
equation
y” + 63/ + 10y = 3cos t. l (b) The graphs on the next page correspond to solutions of three of the following equations.
In each case the initial conditions are y(0) = 2,y'(0) = 0. Indicate which graph
corresponds to which equation; brieﬂy state your reasons for your conclusions. yu+y=0 ____‘
y”+y=t+cos3t ____ y" + y = cos(t/3) ____ l
yII+ZyI+y=0 ___— II 1I _
y +Zy +y—cost __ y" + iy' + y = emu/3) _* ...
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 Spring '04
 Yoon
 Differential Equations, Equations, Boundary value problem, initial conditions, Period __________ Amplitude

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