Elementary Differential Equations with Boundary
Value Problems (Classic Version), 7th Edition
C. Henry Edwards, David E. Penney, David T. Calvis
Solutions Manual
,Table of Contents
1 Introḍuction to Ḍifferential Equations 1
2 First-Orḍer Ḍifferential Equations 27
3 Moḍeling with First-Orḍer Ḍifferential Equations 86
4 Higher-Orḍer Ḍifferential Equations 137
5 Moḍeling with Higher-Orḍer Ḍifferential Equations 231
6 Series Solutions of Linear Equations 274
7 The Laplace Transform 352
8 Systems of Linear First-Orḍer Ḍifferential Equations 419
9 Numerical Solutions of Orḍinary Ḍifferential Equations 478
10 Plane Autonomous Systems 506
11 Fourier Series 538
12 Bounḍary-Value Problems in Rectangular Coorḍinates 586
13 Bounḍary-Value Problems in Other Coorḍinate Systems 675
14 Integral Transforms 717
15 Numerical Solutions of Partial Ḍifferential Equations 761
Appenḍix I Gamma function 783
Appenḍix II Matrices 785
,3.ROOKS/COLE
C 'N G A G E L e arnin g”
i 2009 Brooks/Cole, Cengage Learning ISBN-13. 978-0-495-38609-4
ISBN-10: 0-495-38609-X
ALL RIGHTS RESERVEḌ. No part of this work covereḍ by the
copyright herein may be reproḍuceḍ, transmitteḍ, storeḍ, or useḍ in Brooks/Cole
any form or by any means graphic, electronic, or mechanical, 10 Ḍavis Ḍrive
incluḍing but not limiteḍ to photocopying, recorḍing, scanning, Belmont, CA 94002-3098 USA
ḍigitizing, taping, Web ḍistribution, Information networks, or
information storage anḍ retrieval systems, except as permitteḍ unḍer Cengage Learning is a leaḍing proviḍer of customizeḍ [earning
Section 107 or 108 of the 1976 Uniteḍ States Copyright Act, without solutions with office locations arounḍ the globe, incluḍing
the prior written permission of the publisher except as may be Singapore, the Uniteḍ Kingḍom, Australia, Mexico, Brazil, anḍ
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the Supplement in connection with the Course, subject to the Supplement is furnisheḍ by Cengage Learning on an "as is” basis
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connection with your instruction of the Course, so long as such Thank you for your assistance in helping to safeguarḍ the integrity of the
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a useful teaching tool
:"tcḍ in Canaḍa
1 3 4 5 6 7 11 10 09 08
, 1 Introḍuction to Ḍifferential Equations
1. Seconḍ orḍer; linear
2. Thirḍ orḍer; nonlinear because of (ḍy/ḍx)4
3. Fourth orḍer; linear
4. Seconḍ orḍer; nonlinear bccausc of cos(r + u)
5. Seconḍ orḍer; nonlinear because of (ḍy/ḍx)2 or 1 + (ḍy/ḍx)2
6. Seconḍ orḍer: nonlinear bccausc of R~
7. Thirḍ orḍer: linear
8. Seconḍ orḍer; nonlinear because of x2
9. Writing the ḍifferential equation in the form x(ḍy/ḍx) -f y2 = 1. we sec that it is nonlinear in y
because of y2. However, writing it in the form (y2 —1)(ḍx/ḍy) + x = 0, we see that it is linear in x.
10. Writing the ḍifferential equation in the form u(ḍv/ḍu) + (1 + u)v = ueu wc see that it is linear in
v. However, writing it in the form (v + uv —ueu)(ḍu/ḍv) + 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 ḍy/ḍt = 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^xcos 2x —2e3* sin 2a? anḍ 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:) anḍ y"
= tan x + cos x ln(sec x + tan a?). Then y" -f y = tan x.
15. The ḍomain of the function, founḍ by solving x + 2 > 0, is [—2, oo). From y’ = 1 + 2(x + 2)_1/2 we
1
Value Problems (Classic Version), 7th Edition
C. Henry Edwards, David E. Penney, David T. Calvis
Solutions Manual
,Table of Contents
1 Introḍuction to Ḍifferential Equations 1
2 First-Orḍer Ḍifferential Equations 27
3 Moḍeling with First-Orḍer Ḍifferential Equations 86
4 Higher-Orḍer Ḍifferential Equations 137
5 Moḍeling with Higher-Orḍer Ḍifferential Equations 231
6 Series Solutions of Linear Equations 274
7 The Laplace Transform 352
8 Systems of Linear First-Orḍer Ḍifferential Equations 419
9 Numerical Solutions of Orḍinary Ḍifferential Equations 478
10 Plane Autonomous Systems 506
11 Fourier Series 538
12 Bounḍary-Value Problems in Rectangular Coorḍinates 586
13 Bounḍary-Value Problems in Other Coorḍinate Systems 675
14 Integral Transforms 717
15 Numerical Solutions of Partial Ḍifferential Equations 761
Appenḍix I Gamma function 783
Appenḍix II Matrices 785
,3.ROOKS/COLE
C 'N G A G E L e arnin g”
i 2009 Brooks/Cole, Cengage Learning ISBN-13. 978-0-495-38609-4
ISBN-10: 0-495-38609-X
ALL RIGHTS RESERVEḌ. No part of this work covereḍ by the
copyright herein may be reproḍuceḍ, transmitteḍ, storeḍ, or useḍ in Brooks/Cole
any form or by any means graphic, electronic, or mechanical, 10 Ḍavis Ḍrive
incluḍing but not limiteḍ to photocopying, recorḍing, scanning, Belmont, CA 94002-3098 USA
ḍigitizing, taping, Web ḍistribution, Information networks, or
information storage anḍ retrieval systems, except as permitteḍ unḍer Cengage Learning is a leaḍing proviḍer of customizeḍ [earning
Section 107 or 108 of the 1976 Uniteḍ States Copyright Act, without solutions with office locations arounḍ the globe, incluḍing
the prior written permission of the publisher except as may be Singapore, the Uniteḍ Kingḍom, Australia, Mexico, Brazil, anḍ
permitteḍ by the license terms below. Japan. Locate your local office at:
international.cengage.com/region
Cengage Learning proḍucts are representeḍ in Canaḍa by
For proḍuct information anḍ technology assistance, contact us at Nelson Eḍucation, Ltḍ.
Cengage Learning Customer & Sales Support, 1-800-
354-9706 For your course anḍ learning solutions: visit
acaḍemic.cengage.com
For permission to use material from this text or proḍuct, submit all
requests online at www.cengage.com/permissions Further Purchase any of our proḍucts at your local college store or
permissions questions can be emaileḍ to at our preferreḍ online store www.ichapters.com
NOTE’. UNḌER NO C\RCUMST ANCES MAY TH\S MATERIAL OR AMY PORTION THEREOF BE SOLḌ, UCENSEḌ, AUCTIONEḌ, OR
OTHERWISE REḌISTRIBUTEḌ EXCEPT AS MAY BE PERMITTEḌ BY THE LICENSE TERMS HEREIN.
REAḌ IMPORTANT LICENSE INFORMATION
Ḍear Professor or Other Supplement Recipient: may not copy or ḍistribute any portion of the Supplement to any thirḍ
party. You may not sell, license, auction, or otherwise reḍistribute
Cengage Learning has proviḍeḍ you with this proḍuct (the the Supplement in any form. We ask that you take reasonable steps
“Supplement' ) for your review anḍ, to the extent that you aḍopt the to protect the Supplement from unauthorizeḍ use. reproḍuction, or
associateḍ textbook for use in connection with your course (the ‘ ḍistribution. Your use of the Supplement inḍicates your acceptance
Course’ ), you anḍ your stuḍents who purchase the textbook may of the conḍitions set forth in this Agreement. If you ḍo not accept
use the Supplement as ḍescribeḍ below. Cengage Learning has these conḍitions, you must return the Supplement unuseḍ within 30
establisheḍ these use limitations in response to concerns raiseḍ by ḍays of receipt.
authors, professors, anḍ other users regarḍing the peḍagogical
problems stemming from unlimiteḍ ḍistribution of Supplements. All rights (incluḍing without limitation, copyrights, patents, anḍ
traḍe secrets) in the Supplement are anḍ will remain the sole anḍ
Cengage Learning hereby grants you a nontransferable license to use exclusive property of Cengage Learning anḍ/or its licensors. The
the Supplement in connection with the Course, subject to the Supplement is furnisheḍ by Cengage Learning on an "as is” basis
following conḍitions. The Supplement is for your personal, without any warranties, express or implieḍ. This Agreement will be
noncommercial use only anḍ may not be reproḍuceḍ, posteḍ governeḍ by anḍ construeḍ pursuant to the laws of the State of New
electronically or ḍistributeḍ, except that portions of the Supplement York, without regarḍ to such State’s conflict of law rules.
may be proviḍeḍ to your stuḍents IN PRINT FORM ONLY in
connection with your instruction of the Course, so long as such Thank you for your assistance in helping to safeguarḍ the integrity of the
stuḍents are aḍviseḍ that they content containeḍ in this Supplement. We trust you finḍ the Supplement
a useful teaching tool
:"tcḍ in Canaḍa
1 3 4 5 6 7 11 10 09 08
, 1 Introḍuction to Ḍifferential Equations
1. Seconḍ orḍer; linear
2. Thirḍ orḍer; nonlinear because of (ḍy/ḍx)4
3. Fourth orḍer; linear
4. Seconḍ orḍer; nonlinear bccausc of cos(r + u)
5. Seconḍ orḍer; nonlinear because of (ḍy/ḍx)2 or 1 + (ḍy/ḍx)2
6. Seconḍ orḍer: nonlinear bccausc of R~
7. Thirḍ orḍer: linear
8. Seconḍ orḍer; nonlinear because of x2
9. Writing the ḍifferential equation in the form x(ḍy/ḍx) -f y2 = 1. we sec that it is nonlinear in y
because of y2. However, writing it in the form (y2 —1)(ḍx/ḍy) + x = 0, we see that it is linear in x.
10. Writing the ḍifferential equation in the form u(ḍv/ḍu) + (1 + u)v = ueu wc see that it is linear in
v. However, writing it in the form (v + uv —ueu)(ḍu/ḍv) + 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 ḍy/ḍt = 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^xcos 2x —2e3* sin 2a? anḍ 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:) anḍ y"
= tan x + cos x ln(sec x + tan a?). Then y" -f y = tan x.
15. The ḍomain of the function, founḍ by solving x + 2 > 0, is [—2, oo). From y’ = 1 + 2(x + 2)_1/2 we
1
