Complete Solutions Manual
A First Course in Differential Equations
with Modeling Applications
Ninth Edition
Dennis G. Zill
Loyola Marymount University
Differential Equations with
Boundary-Vary Problems
Seventh Edition
Dennis G. Zill
Loyola Marymount University
Michael R. Cullen
Late of Loyola Marymount University
By
Warren S. Wright
Loyola Marymount University
Carol D. Wright
* ; BROOKS/COLE
C E N G A G E Learning-
Australia • Brazil - Japan - Korea • Mexico • Singapore • Spain • United Kingdom • United States
, Table of Contents
1 Introduction to Differential Equations 1
2 First-Order Differential Equations 27
3 Modeling with First-Order Differential Equations 86
4 Higher-Order Differential Equations 137
5 Modeling with Higher-Order Differential Equations 231
6 Series Solutions of Linear Equations 274
7 The Laplace Transform 352
8 Systems of Linear First-Order Differential Equations 419
9 Numerical Solutions of Ordinary Differential Equations 478
10 Plane Autonomous Systems 506
11 Fourier Series 538
12 Boundary-Value Problems in Rectangular Coordinates 586
13 Boundary-Value Problems in Other Coordinate Systems 675
14 Integral Transforms 717
15 Numerical Solutions of Partial Differential Equations 761
Appendix I Gamma function 783
A ppendix II Matrices 785
,3.ROOKS/COLE
C 'N G A G E L e a rn in g ”
i 2009 Brooks/Cole, Cengage Learning ISBN-13. 978-0-495-38609-4
ISBN-10: 0-495-38609-X
ALL RIGHTS RESERVED. No part of this work covered by the
copyright herein may be reproduced, transmitted, stored, or Brooks/Cole
used in any form or by any means graphic, electronic, or 10 Davis Drive
mechanical, including but not limited to photocopying, Belmont, CA 94002-3098
recording, scanning, digitizing, taping, Web distribution, USA
Information networks, or information storage and retrieval
systems, except as permitted under Section 107 or 108 of the Cengage Learning is a leading provider of customized
1976 United States Copyright Act, without the prior written [earning solutions with office locations around the globe,
permission of the publisher except as may be permitted by the including Singapore, the United Kingdom, Australia,
license terms below. Mexico, Brazil, and Japan. Locate your local office at:
international.cengage.com/region
Cengage Learning products are represented in
For product information and technology assistance, contact us at Canada by Nelson Education, Ltd.
Cengage Learning Customer & Sales Support,
1-800-354-9706 For your course and learning solutions: visit
academic.cengage.com
For permission to use material from this text or product, submit
all requests online at www.cengage.com/permissions Purchase any of our products at your local college
Further permissions questions can be emailed to store or at our preferred online store
permissionreq uest@cenga ge. com
www.ichapters.com
NOTE’. UNDER NO C\RCUMST ANCES MAY TH\S MATERIAL OR AMY PORTION THEREOF BE SOLD, UCENSED, AUCTIONED,
OR OTHERWISE REDISTRIBUTED EXCEPT AS MAY BE PERMITTED BY THE LICENSE TERMS HEREIN.
READ IMPORTANT LICENSE INFORMATION
Dear Professor or Other Supplement Recipient: may not copy or distribute any portion of the Supplement to any
third party. You may not sell, license, auction, or otherwise
Cengage Learning has provided you with this product (the redistribute the Supplement in any form. We ask that you take
“Supplement' ) for your review and, to the extent that you adopt reasonable steps to protect the Supplement from unauthorized
the associated textbook for use in connection with your course use. reproduction, or distribution. Your use of the Supplement
(the ‘Course’ ), you and your students who purchase the indicates your acceptance of the conditions set forth in this
textbook may use the Supplement as described below. Cengage Agreement. If you do not accept these conditions, you must
Learning has established these use limitations in response to return the Supplement unused within 30 days of receipt.
concerns raised by authors, professors, and other users
regarding the pedagogical problems stemming from unlimited All rights (including without limitation, copyrights, patents, and
distribution of Supplements. trade secrets) in the Supplement are and will remain the sole and
exclusive property of Cengage Learning and/or its licensors. The
Cengage Learning hereby grants you a nontransferable license Supplement is furnished by Cengage Learning on an "as is” basis
to use the Supplement in connection with the Course, subject to without any warranties, express or implied. This Agreement will
the following conditions. The Supplement is for your personal, be governed by and construed pursuant to the laws of the State
noncommercial use only and may not be reproduced, posted of New York, without regard to such State’s conflict of law rules.
electronically or distributed, except that portions of the
Supplement may be provided to your students IN PRINT FORM Thank you for your assistance in helping to safeguard the integrity
ONLY in connection with your instruction of the Course, so long of the content contained in this Supplement. We trust you find the
as such students are advised that they Supplement a useful teaching tool
:"tcd in Canada
1 3 4 5 6 7 11 10 09 08
, 1 Introduction to Differential Equations
1 . Second order; linear
2 . Third order; nonlinear because of (dy/dx)4
3. Fourth order; linear
4. Second order; nonlinear bccausc of cos(r + u)
5. Second order; nonlinear because of (dy/dx)2 or 1 + (dy/dx)2
6 . Second order: nonlinear bccausc of R~
7. Third order: linear
8 . Second order; nonlinear because of x2
9. Writing the differential equation in the form x(dy/dx) -f y2 = 1. we sec that it is nonlinear in y
because of y2. However, writing it in the form (y2 —1)(dx/dy) + x = 0, we see that it is linear in x.
10. Writing the differential equation in the form u(dv/du) + (1 + u)v = ueu wc see that it is linear in
v. However, writing it in the form (v + uv —ueu)(du/dv) + u — 0, we see that it, is nonlinear in ■
Ji
ll. From y = e-*/2 we obtain y' = —\e~x'2. Then 2y' + y = —e~X//2 + e-x/2 = 0.
12 . From y = | — |e-20* we obtain dy/dt = 24e-20t, so that
% + 20y = 24e~m + 20 - |e_20t) = 24.
clt \'o 5 /
13. R'om y = eix cos 2x we obtain y1= 3e^x cos 2x — 2e3* sin 2a? and y” = 5e3,xcos 2x — 12e3,xsin 2x, so
that y" — (k/ + l?>y = 0.
14. From y = —cos:r ln(sec;r + tanrc) we obtain y’ — —1 + sin.Tln(secx + tana:) and
y" = tan x + cos x ln(sec x + tan a?). Then y" -f y = tan x.
15. The domain of the function, found by solving x + 2 > 0, is [—2, oo). From y’ = 1 + 2(x + 2)_1/2 we
1
A First Course in Differential Equations
with Modeling Applications
Ninth Edition
Dennis G. Zill
Loyola Marymount University
Differential Equations with
Boundary-Vary Problems
Seventh Edition
Dennis G. Zill
Loyola Marymount University
Michael R. Cullen
Late of Loyola Marymount University
By
Warren S. Wright
Loyola Marymount University
Carol D. Wright
* ; BROOKS/COLE
C E N G A G E Learning-
Australia • Brazil - Japan - Korea • Mexico • Singapore • Spain • United Kingdom • United States
, Table of Contents
1 Introduction to Differential Equations 1
2 First-Order Differential Equations 27
3 Modeling with First-Order Differential Equations 86
4 Higher-Order Differential Equations 137
5 Modeling with Higher-Order Differential Equations 231
6 Series Solutions of Linear Equations 274
7 The Laplace Transform 352
8 Systems of Linear First-Order Differential Equations 419
9 Numerical Solutions of Ordinary Differential Equations 478
10 Plane Autonomous Systems 506
11 Fourier Series 538
12 Boundary-Value Problems in Rectangular Coordinates 586
13 Boundary-Value Problems in Other Coordinate Systems 675
14 Integral Transforms 717
15 Numerical Solutions of Partial Differential Equations 761
Appendix I Gamma function 783
A ppendix II Matrices 785
,3.ROOKS/COLE
C 'N G A G E L e a rn in g ”
i 2009 Brooks/Cole, Cengage Learning ISBN-13. 978-0-495-38609-4
ISBN-10: 0-495-38609-X
ALL RIGHTS RESERVED. No part of this work covered by the
copyright herein may be reproduced, transmitted, stored, or Brooks/Cole
used in any form or by any means graphic, electronic, or 10 Davis Drive
mechanical, including but not limited to photocopying, Belmont, CA 94002-3098
recording, scanning, digitizing, taping, Web distribution, USA
Information networks, or information storage and retrieval
systems, except as permitted under Section 107 or 108 of the Cengage Learning is a leading provider of customized
1976 United States Copyright Act, without the prior written [earning solutions with office locations around the globe,
permission of the publisher except as may be permitted by the including Singapore, the United Kingdom, Australia,
license terms below. Mexico, Brazil, and Japan. Locate your local office at:
international.cengage.com/region
Cengage Learning products are represented in
For product information and technology assistance, contact us at Canada by Nelson Education, Ltd.
Cengage Learning Customer & Sales Support,
1-800-354-9706 For your course and learning solutions: visit
academic.cengage.com
For permission to use material from this text or product, submit
all requests online at www.cengage.com/permissions Purchase any of our products at your local college
Further permissions questions can be emailed to store or at our preferred online store
permissionreq uest@cenga ge. com
www.ichapters.com
NOTE’. UNDER NO C\RCUMST ANCES MAY TH\S MATERIAL OR AMY PORTION THEREOF BE SOLD, UCENSED, AUCTIONED,
OR OTHERWISE REDISTRIBUTED EXCEPT AS MAY BE PERMITTED BY THE LICENSE TERMS HEREIN.
READ IMPORTANT LICENSE INFORMATION
Dear Professor or Other Supplement Recipient: may not copy or distribute any portion of the Supplement to any
third party. You may not sell, license, auction, or otherwise
Cengage Learning has provided you with this product (the redistribute the Supplement in any form. We ask that you take
“Supplement' ) for your review and, to the extent that you adopt reasonable steps to protect the Supplement from unauthorized
the associated textbook for use in connection with your course use. reproduction, or distribution. Your use of the Supplement
(the ‘Course’ ), you and your students who purchase the indicates your acceptance of the conditions set forth in this
textbook may use the Supplement as described below. Cengage Agreement. If you do not accept these conditions, you must
Learning has established these use limitations in response to return the Supplement unused within 30 days of receipt.
concerns raised by authors, professors, and other users
regarding the pedagogical problems stemming from unlimited All rights (including without limitation, copyrights, patents, and
distribution of Supplements. trade secrets) in the Supplement are and will remain the sole and
exclusive property of Cengage Learning and/or its licensors. The
Cengage Learning hereby grants you a nontransferable license Supplement is furnished by Cengage Learning on an "as is” basis
to use the Supplement in connection with the Course, subject to without any warranties, express or implied. This Agreement will
the following conditions. The Supplement is for your personal, be governed by and construed pursuant to the laws of the State
noncommercial use only and may not be reproduced, posted of New York, without regard to such State’s conflict of law rules.
electronically or distributed, except that portions of the
Supplement may be provided to your students IN PRINT FORM Thank you for your assistance in helping to safeguard the integrity
ONLY in connection with your instruction of the Course, so long of the content contained in this Supplement. We trust you find the
as such students are advised that they Supplement a useful teaching tool
:"tcd in Canada
1 3 4 5 6 7 11 10 09 08
, 1 Introduction to Differential Equations
1 . Second order; linear
2 . Third order; nonlinear because of (dy/dx)4
3. Fourth order; linear
4. Second order; nonlinear bccausc of cos(r + u)
5. Second order; nonlinear because of (dy/dx)2 or 1 + (dy/dx)2
6 . Second order: nonlinear bccausc of R~
7. Third order: linear
8 . Second order; nonlinear because of x2
9. Writing the differential equation in the form x(dy/dx) -f y2 = 1. we sec that it is nonlinear in y
because of y2. However, writing it in the form (y2 —1)(dx/dy) + x = 0, we see that it is linear in x.
10. Writing the differential equation in the form u(dv/du) + (1 + u)v = ueu wc see that it is linear in
v. However, writing it in the form (v + uv —ueu)(du/dv) + u — 0, we see that it, is nonlinear in ■
Ji
ll. From y = e-*/2 we obtain y' = —\e~x'2. Then 2y' + y = —e~X//2 + e-x/2 = 0.
12 . From y = | — |e-20* we obtain dy/dt = 24e-20t, so that
% + 20y = 24e~m + 20 - |e_20t) = 24.
clt \'o 5 /
13. R'om y = eix cos 2x we obtain y1= 3e^x cos 2x — 2e3* sin 2a? and y” = 5e3,xcos 2x — 12e3,xsin 2x, so
that y" — (k/ + l?>y = 0.
14. From y = —cos:r ln(sec;r + tanrc) we obtain y’ — —1 + sin.Tln(secx + tana:) and
y" = tan x + cos x ln(sec x + tan a?). Then y" -f y = tan x.
15. The domain of the function, found by solving x + 2 > 0, is [—2, oo). From y’ = 1 + 2(x + 2)_1/2 we
1
