Exam2Fall94 - 65.2400 — 20, 21, 22 TEST 2 Nov. 2, 1994...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

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 spring-mass 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 y-axis. (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 = ...
View Full Document

Page1 / 6

Exam2Fall94 - 65.2400 — 20, 21, 22 TEST 2 Nov. 2, 1994...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online