Exam3Spring96 - 65 .2405 TEST 3 April 24, 1996 Name 1....

Info iconThis preview shows pages 1–9. 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
Background image of page 7

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

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 65 .2405 TEST 3 April 24, 1996 Name 1. Please answer all questions, showing your work in detail and giving reasons where appropriate. 2. This is an open book test. Collaboration with other students is NOT permitted. 3. Point allocations for each question are indicated. Plan your time accordingly. 4. In some questions it is specified that you solve the problem by hand. In all other questions you are encouraged to use Maple whenever you think it will be helpful. 5. Be sure that you have all 8 test pages in addition to this cover sheet. NAME ___. ____..______________ 1. The eigenvalues and eigenvectors of a certain matrix A are 7'1: ’17 gm = T2 = —1/27 5(2) = (a) (6 points) Sketch a phase portrait for the system 2’ = Ax. x7. (b) (6 points) Find the solution of the initial value problem NAME (C) (4 points) Sketch the graphs of 171 versus t and $2 versus t for the solution in part (d) (4 points) Classify the point x = 0 as to type (saddle point, node, spiral point, center) and state whether it is asymptotically stable, stable, or unstable. NAME __ _________. _.______.. 2. (a) (10 points) Find by hand the eigenvalues and eigenvectors of the matrix B=(:§1‘)- (b) (10 points) The eigenvalues and eigenvectors of a certain matrix A are 3 3 r1 + 2’5 <2+z>’ T2 ’é 2—2“ Find the general solution, using real-valued functions, of the system I’ = Ax. NAME ____. ______________ __ 3. (a) (5 points) A phase portrait for a certain system 1’ = A3: is shown below. . What can you say about the eigenvalues and eigenvectors of the matrix A? (b) (5 points) A phase portrait for a certain system 1" = A3: is shown below. Sketch graphs of 1:1 versus t and $2 versus t that are consistent with this phase plot. Be sure to identify which graph is 1:1 and which is 2:2. X1 Y (3,» NAME .______ (c) (5 points) One of the systems listed below has the phase portrait that is shown below. Mark clearly the system having this phase portrait, and explain how you reached your conclusion. ‘ NAME ____________________ 4. Consider the system x’ = 1 + 2y, y’ = l — 3x2 (a) (4 points) Find all of the equilibrium solutions (critical points). (b) (10 points) Choose a rectangle that includes all of the critical points. Construct a phase portrait for this syitem. Sketch your phase portrait below. *2 (c) (6 points) From your phase portrait classify each critical point as to type (saddle point, node, spiral point, center) and state whether it is asymptotically stable, stable, or unstable. NAME _____________.. __ 5. The functions u (a: t) - sin m”: cos mm (1) ’f ’ _ 10 10 satisfy the wave equation U11» = U“, 0 < :1: < 10, t> 0 the boundary conditions u(0, t) = O, u(10, t) = 0, t > 0 (3) and the initial condition ut(:r,0) = 0, 0 < :r < 10 (4) for all positive integers n. (a) (8 points) Suppose that we want to find the function u(3:, t) that satisfies (2), (3), and (4) and also the second initial condition u(:r, 0) = x(10 — z)2/125, 0 < a: <10 Write down the form of an infinite series expression for u(a:, t), and provide an integral formula for the coefficients. (b) (7 points) Evaluate the coefficients from part (a). NAME _._____________.____ (c) (5 points) Plot u(3,t) versus t for at least two periods, using a moderate number of terms in the series. Sketch the graph of u(3, t) versus t below. u(3,t\ (d) (5 points) Plot u versus :2: for 0 g a: S 10 and for t = 7. Sketch the graph below. uhfl) lO ...
View Full Document

This test prep was uploaded on 04/09/2008 for the course MATH 2400 taught by Professor Yoon during the Spring '04 term at Rensselaer Polytechnic Institute.

Page1 / 9

Exam3Spring96 - 65 .2405 TEST 3 April 24, 1996 Name 1....

This preview shows document pages 1 - 9. Sign up to view the full document.

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