Differential Equations And Boundary Value Problems – C Henry Edwards

For instance, suppose that x1.0/ D x2.0/ D 0 and that x0 1.0/ D x0 2.0/ D v0. Then the equations x1.0/ D c1 C c2 C x2.0/ D c1 C c2   2c2  2c3 C c4 D v0,  2c2 C 2c3  c4 D v0 are readily solved for c1 D 1 2v0, c2 D  1 2v0, and c3 D c4 D 0, so x1.t/ D x2.t/ D 1  1  e2t ; 1.t/ D x0 2.t/ D v0e2t: In this case the two railway cars continue in the same direction with equal but exponentially damped velocities, approaching the displacements x1 D x2 D 1 2v0 as t !

C1. It is of interest to interpret physically the individual generalized eigenvector solutions given in (33). The degenerate (0 D 0) solution x1.t/ D describes the two masses at rest with position functions x1.t/  1 and x2.t/  1. The solution x2.t/ D 2 2 T e2t corresponding to the carefully chosen eigenvector w1 describes damped motions x1.t/ D e2t and x2.t/ D e2t of the two masses, with equal velocities in the same direction.

Finally, the solutions x3.t/ and x4.t/ resulting from the length 2 chain fv1; v2g both describe damped motion with the two masses moving in opposite directions. The methods of this section apply to complex multiple eigenvalues just as to real multiple eigenvalues (although the necessary computations tend to be somewhat lengthy). Given a complex conjugate pair ˛ ˙ ˇi of eigenvalues of multiplicity k, we work with one of them (say, ˛  ˇi) as if it were real to find k independent complex-valued solutions.

The real and imaginary parts of these complex-valued solutions then provide 2k real-valued solutions associated with the two eigenvalues  D ˛  ˇi and  D ˛ C ˇi each of multiplicity k. See Problems 33 and 34. 5.5 Problems Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graph- ing calculator to construct a direction field and typical solution curves for the given system.

1. x0 D  2 1 4  2. x0 D  3 1  3. x0 D  1 2  4. x0 D  3 1  5.5 Multiple Eigenvalue Solutions 5. x0 D  4  6. x0 D  1 4  7. x0 D 7 5 x 8. x0 D 18 5 5 x 9. x0 D 19 8 5 x 10. x0 D 13 48 8 24 5 x 11.

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H. (Charles Henry) Differential equations and boundary value problems : computing and modeling / C. Henry Edwards, David E. Penney, The University of Georgia, David Calvis, Baldwin Wallace College. — Fifth edition. pages cm ISBN 978-0-321-79698-1 (hardcover) 1. Differential equations. 2. Boundary value problems. I. Penney, David E. II. Calvis, David. III. Title. QA371.E28 2015 515′.35–dc23 2013040067 Copyright c 2015, 2008, 2004 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or trans- mitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

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