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Unformatted text preview: 65.2400 — 20, 21, 22 TEST 2 Nov. 2, 1994 Name 'Section 20 21 22 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. NAME ____________________________ SECTION ____________________ __ 1. (a) (15 points) The position of a certain springmass system satisﬁes the differential
equation
y” + 2y’ + 5y = 35m 2t. Find the steady (long—term) motion of this system. What is the amplitude of this
steady motion? 1. (b) (10 points) The graph of the position u of a certain spring—mass system as a function
of time t is shown below. On the other set of axes sketch the corresponding parametric
graph of 11’ versus u. NAME _______________________________ __ SECTION _____________ _____ 2. (a) (10 points) Find the solution of the initial value problem 1/” + 62/ + 9y = 0, y(0) = 2, y’(0) = 1 (10 points) Find the steady state temperature distribution in a bar subject to the
' boundary conditions u(0, t) = 50, ux(20,t) = —2. NAME _________________________ ____ SECTION ________________ __ / 3. (a) (10 points) Suppose that Where 1 5
bn=—/(5—x)sinﬂr£,dx, n=1,2,....
5 o 5 On the axes below draw the graph of y = f(x) for —10 < x < 10. I 3. (b) (15 points) Consider the problem
X”+aX=0, X(0)=0, X’(12)=0, Where 0' is a positive constant. Find the eigenvalues and eigenfunctions, that is, ﬁnd the values of a for which nonzero solutions X exist and also find the corresponding
nonzero solutions X LAB TEST 2 65.2400  20, 21, 22 November 4, 1994 NAME SECTION
4. Let f(x) = (a:  5)2/10 for O S :r S 5. The Fourier cosine series for this function has the form °° km:
+ Z (1;. cos —.
k=1 5 (10
2 (a) Find an expression for the general coefﬁcient ak and write it in the space below. (b) Evaluate a3 and am as ﬂoating point (decimal) numbers. <13=—————— aio= (e) Let Sn(2:) be the partial sum including only terms up through an cos (mrx/ 5). Plot
f (2:) — 510(2) for 0 g :r g 5 and sketch the graph below. Be sure to put a scale on
the yaxis. (d) Where is the value of the error — 510(x) greatest? (e) Find the smallest value of n for which the error f(1:) — Sn(a:)[ is less than 0.05 for all
:r in 0 S :c S 5. 5. A certain vibrating system satisﬁes the initial value problem u" + gu’ + 3n = O, u(0) = 1, u’(0) = 2. v (a) Use dsolve to ﬁnd the solution of this problem and write it below. (b) Plot u versus t for the solution you found in (a). Sketch its graph on the axes below.
Be sure to put scales on your axes. (L (0) Estimate from your graph the maximum value achieved by u. umax (d) Find, with at least three decimal place accuracy, the time T for which u(t) < 0.01
for all t > T.
T = ...
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 Spring '04
 Yoon
 Differential Equations, Equations, Fourier Series, Boundary value problem

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