SOLUTION MANUAL
DIFFERENTIAL EQUATIONS WITH BOUNDARY
VALUE PROBLEMS
, Complete Solutions Manual
A First Course In Differential Equations
With Modeling Applications
Eleventh Edition
Dennis G. Zill
Loyola Marymount University
Differential Equations
With Boundary-Vary
Problems
Tenth 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
Appendix 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 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 Cengage Learning Is A Leading Provider Of Customized
The 1976 United States Copyright Act, Without The Prior [Earning Solutions With Office Locations Around The
Written Permission Of The Publisher Except As May Be Globe, Including Singapore, The United Kingdom,
Permitted By The License Terms Below. Australia, 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 Canada By Nelson Education, Ltd.
At
Cengage Learning Customer & Sales Support, For Your Course And Learning Solutions: Visit
1-800-354-9706 Academic.Cengage.Com
For Permission To Use Material From This Text Or Product, Purchase Any Of Our Products At Your Local College
Submit All Requests Online At Store Or At Our Preferred Online Store
Www.Cengage.Com/Permissions Further Permissions Www.Ichapters.Com
Questions Can Be Emailed To
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
Cengage Learning Has Provided You With This Product Otherwise Redistribute The Supplement In Any Form. We Ask
(The “Supplement' ) For Your Review And, To The Extent That That You Take Reasonable Steps To Protect The Supplement
You Adopt The Associated Textbook For Use In Connection From Unauthorized Use. Reproduction, Or Distribution. Your
With Your Course (The „ Course‟ ), You And Your Students Use Of The Supplement Indicates Your Acceptance Of The
Who Purchase The Textbook May Use The Supplement As Conditions Set Forth In This Agreement. If You Do Not
Described Below. Cengage Learning Has Established These Accept These Conditions, You Must Return The Supplement
Use Limitations In Response To Concerns Raised By Unused Within 30 Days Of Receipt.
Authors, Professors, And Other Users Regarding The
Pedagogical Problems Stemming From Unlimited Distribution All Rights (Including Without Limitation, Copyrights, Patents,
Of Supplements. And Trade Secrets) In The Supplement Are And Will Remain
The Sole And Exclusive Property Of Cengage Learning And/Or
Cengage Learning Hereby Grants You A Nontransferable Its Licensors. The Supplement Is Furnished By Cengage
License To Use The Supplement In Connection With The Learning On An "As Is” Basis Without Any Warranties, Express
Course, Subject To The Following Conditions. The Or Implied. This Agreement Will Be Governed By And
Supplement Is For Your Personal, Noncommercial Use Only Construed Pursuant To The Laws Of The State Of New York,
And May Not Be Reproduced, Posted Electronically Or Without Regard To Such State‟s Conflict Of Law Rules.
Distributed, Except That Portions Of The Supplement May Be
Provided To Your Students In Print Form Only In Connection Thank You For Your Assistance In Helping To Safeguard The
With Your Instruction Of The Course, So Long As Such Integrity Of The Content Contained In This Supplement. We Trust
Students Are Advised That They You Find The 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
, Definitions and Terminology
Have
{Y - X)Y' = (Y - ®)[I + (20 + 2)_1/2]
= Y —X + 2(Y - X)(X + 2)-1/2
= Y - X + 2[X + 4(Z + 2)1/2 - A;](A: + 2)_1/2
= Y —X + 8(Ac + 2)1;/I(Rr + 2)~1/2 = Y —X + 8.
An Interval Of Definition For The Solution Of The Differential Equation Is (—2, Oo) Because Y
Defined At X = —2.
16. Since Tan:R Is Not Defined For X = 7r/2 + M R, N An Integer, The Domain Of Y =
5t£V.
{A; |5x ^ Tt/2 + 7i-7r} Or {;R |X ^ Tt/Io + Mr/5}. From Y' — 25sec2 §X We Have
Y' = 25(1 + Tan2 5x) = 25 + 25 Tan2 5a: = 25 + Y2.
An Interval Of Definition For The Solution Of The Differential Equation Is (—7r/10, 7t/10 . A:,
Interval Is (7r/10, 37t/10). And So On.
17. The Domain Of The Function Is {X \4 —X2 ^ 0} Or {X\X ^ —2 Or X ^ 2}. Prom Y' —
2.: -=-
We Have
An Interval Of Definition For The Solution Of The Differential Equation Is (—2, 2).
Other (—O C,—2) And (2, Oo).
18. The Function Is Y — L/ Y/ L —S In S . Whose Domain Is Obtained From 1 —Sinx ^ 0 Or .
= 1 T The Domain Is {Z |X ^ Tt/2 + 2?I7r}. From Y' = —1(1 —Sin X) 2(—
Cos.X) We Have
2y' = (1 —Sin;R)_ „?/‟2 Cos# = [(1 —Sin:R)~1/2]3cos:R - F/3 Cosx.
An Interval Of Definition For The Solution Of The Differential Equation Is (Tt/2. 5tt/2
A:.. .
Is (57r/2, 97r/2) And So On.
19. Writing Ln(2x — 1) —Ln(X — 1) = T And Differentiating Implicitly We Obtain
2DX 1 D
X 2X-1 Dt X
- L Dt
2x - 2 - 2x + 1 D X _
(2x - 1)(X - 1) Dt
,Definitions and Terminology
Ix
— = -C2x - 1)(X - 1) = (X - 1)(1 - 2x .
2
, Definitions and Terminology
Exponentiating Both Sides Of The Implicit Solution We Obtain X
2X-1
----- = El
X - L
2 X - 1 = Xel - Ef
-4 -2
(E* - 1) = (E„ - 2)X -2
Ef' — 1
X = -4
E* - 2 '
Solving E* — 2 = 0 We Get T = In 2. Thus, The Solution Is Defined On (—Oc.Ln2) Or On (In2,
Oo). The Graph Of The Solution Defined On (—Oo,Ln2) Is Dashed, And The Graph Of The Solution
Defined On (In 2. Oc) Is Solid.
20. Implicitly Differentiating The Solution, We Obtain Y
—X2dy —2xy Dx + Y Dy — 0
2xy Dx + (A;2 —Y)Dy = 0.
Using The Quadratic Formula To Solve Y2—2x2y —1 = 0 For Y, We Get
Y = (2x2 ± V4;C4 + 4)/2 = A‟2 ± V V 1+ 1 . Thus, Two Explicit
Solutions Are Y\ = X2 + \A'4 + 1 And Y-2 = X2 — V.X4 + 1.
Both Solutions Are Defined On (—Oo. Oc). The Graph Of Yj (X) Is
Solid And The Graph Of Y-2 Is Daalied.
21. Differentiating P = C\?}} ( L + Cie^ We Obtain
Dp _ ( L + Cie*) Cie* - Cie* • Cie* _ Cie« [(L + Cie„) - Cie4]
Eft (1 + Cie*)" 1+ 1 + Cie(
Cief
Ci Cic
1 + Ci&- 1 + Ci
= P( 1 - P).
Ef
2 Px ,2 ,2
22. Differentiating Y = E~X / E: Dt + C\E~X We Obtain
Jo
/ = E- 2xe 2x* F *22 J -R.2 2 Rx +2 X
*2er2 =
r
Y X 2xe
/ E1 Dt — X E Dt —
2c\Xe 2cixe Jo
Jo
Substituting Into The Differential Equation, We
Have
DIFFERENTIAL EQUATIONS WITH BOUNDARY
VALUE PROBLEMS
, Complete Solutions Manual
A First Course In Differential Equations
With Modeling Applications
Eleventh Edition
Dennis G. Zill
Loyola Marymount University
Differential Equations
With Boundary-Vary
Problems
Tenth 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
Appendix 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 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 Cengage Learning Is A Leading Provider Of Customized
The 1976 United States Copyright Act, Without The Prior [Earning Solutions With Office Locations Around The
Written Permission Of The Publisher Except As May Be Globe, Including Singapore, The United Kingdom,
Permitted By The License Terms Below. Australia, 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 Canada By Nelson Education, Ltd.
At
Cengage Learning Customer & Sales Support, For Your Course And Learning Solutions: Visit
1-800-354-9706 Academic.Cengage.Com
For Permission To Use Material From This Text Or Product, Purchase Any Of Our Products At Your Local College
Submit All Requests Online At Store Or At Our Preferred Online Store
Www.Cengage.Com/Permissions Further Permissions Www.Ichapters.Com
Questions Can Be Emailed To
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
Cengage Learning Has Provided You With This Product Otherwise Redistribute The Supplement In Any Form. We Ask
(The “Supplement' ) For Your Review And, To The Extent That That You Take Reasonable Steps To Protect The Supplement
You Adopt The Associated Textbook For Use In Connection From Unauthorized Use. Reproduction, Or Distribution. Your
With Your Course (The „ Course‟ ), You And Your Students Use Of The Supplement Indicates Your Acceptance Of The
Who Purchase The Textbook May Use The Supplement As Conditions Set Forth In This Agreement. If You Do Not
Described Below. Cengage Learning Has Established These Accept These Conditions, You Must Return The Supplement
Use Limitations In Response To Concerns Raised By Unused Within 30 Days Of Receipt.
Authors, Professors, And Other Users Regarding The
Pedagogical Problems Stemming From Unlimited Distribution All Rights (Including Without Limitation, Copyrights, Patents,
Of Supplements. And Trade Secrets) In The Supplement Are And Will Remain
The Sole And Exclusive Property Of Cengage Learning And/Or
Cengage Learning Hereby Grants You A Nontransferable Its Licensors. The Supplement Is Furnished By Cengage
License To Use The Supplement In Connection With The Learning On An "As Is” Basis Without Any Warranties, Express
Course, Subject To The Following Conditions. The Or Implied. This Agreement Will Be Governed By And
Supplement Is For Your Personal, Noncommercial Use Only Construed Pursuant To The Laws Of The State Of New York,
And May Not Be Reproduced, Posted Electronically Or Without Regard To Such State‟s Conflict Of Law Rules.
Distributed, Except That Portions Of The Supplement May Be
Provided To Your Students In Print Form Only In Connection Thank You For Your Assistance In Helping To Safeguard The
With Your Instruction Of The Course, So Long As Such Integrity Of The Content Contained In This Supplement. We Trust
Students Are Advised That They You Find The 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
, Definitions and Terminology
Have
{Y - X)Y' = (Y - ®)[I + (20 + 2)_1/2]
= Y —X + 2(Y - X)(X + 2)-1/2
= Y - X + 2[X + 4(Z + 2)1/2 - A;](A: + 2)_1/2
= Y —X + 8(Ac + 2)1;/I(Rr + 2)~1/2 = Y —X + 8.
An Interval Of Definition For The Solution Of The Differential Equation Is (—2, Oo) Because Y
Defined At X = —2.
16. Since Tan:R Is Not Defined For X = 7r/2 + M R, N An Integer, The Domain Of Y =
5t£V.
{A; |5x ^ Tt/2 + 7i-7r} Or {;R |X ^ Tt/Io + Mr/5}. From Y' — 25sec2 §X We Have
Y' = 25(1 + Tan2 5x) = 25 + 25 Tan2 5a: = 25 + Y2.
An Interval Of Definition For The Solution Of The Differential Equation Is (—7r/10, 7t/10 . A:,
Interval Is (7r/10, 37t/10). And So On.
17. The Domain Of The Function Is {X \4 —X2 ^ 0} Or {X\X ^ —2 Or X ^ 2}. Prom Y' —
2.: -=-
We Have
An Interval Of Definition For The Solution Of The Differential Equation Is (—2, 2).
Other (—O C,—2) And (2, Oo).
18. The Function Is Y — L/ Y/ L —S In S . Whose Domain Is Obtained From 1 —Sinx ^ 0 Or .
= 1 T The Domain Is {Z |X ^ Tt/2 + 2?I7r}. From Y' = —1(1 —Sin X) 2(—
Cos.X) We Have
2y' = (1 —Sin;R)_ „?/‟2 Cos# = [(1 —Sin:R)~1/2]3cos:R - F/3 Cosx.
An Interval Of Definition For The Solution Of The Differential Equation Is (Tt/2. 5tt/2
A:.. .
Is (57r/2, 97r/2) And So On.
19. Writing Ln(2x — 1) —Ln(X — 1) = T And Differentiating Implicitly We Obtain
2DX 1 D
X 2X-1 Dt X
- L Dt
2x - 2 - 2x + 1 D X _
(2x - 1)(X - 1) Dt
,Definitions and Terminology
Ix
— = -C2x - 1)(X - 1) = (X - 1)(1 - 2x .
2
, Definitions and Terminology
Exponentiating Both Sides Of The Implicit Solution We Obtain X
2X-1
----- = El
X - L
2 X - 1 = Xel - Ef
-4 -2
(E* - 1) = (E„ - 2)X -2
Ef' — 1
X = -4
E* - 2 '
Solving E* — 2 = 0 We Get T = In 2. Thus, The Solution Is Defined On (—Oc.Ln2) Or On (In2,
Oo). The Graph Of The Solution Defined On (—Oo,Ln2) Is Dashed, And The Graph Of The Solution
Defined On (In 2. Oc) Is Solid.
20. Implicitly Differentiating The Solution, We Obtain Y
—X2dy —2xy Dx + Y Dy — 0
2xy Dx + (A;2 —Y)Dy = 0.
Using The Quadratic Formula To Solve Y2—2x2y —1 = 0 For Y, We Get
Y = (2x2 ± V4;C4 + 4)/2 = A‟2 ± V V 1+ 1 . Thus, Two Explicit
Solutions Are Y\ = X2 + \A'4 + 1 And Y-2 = X2 — V.X4 + 1.
Both Solutions Are Defined On (—Oo. Oc). The Graph Of Yj (X) Is
Solid And The Graph Of Y-2 Is Daalied.
21. Differentiating P = C\?}} ( L + Cie^ We Obtain
Dp _ ( L + Cie*) Cie* - Cie* • Cie* _ Cie« [(L + Cie„) - Cie4]
Eft (1 + Cie*)" 1+ 1 + Cie(
Cief
Ci Cic
1 + Ci&- 1 + Ci
= P( 1 - P).
Ef
2 Px ,2 ,2
22. Differentiating Y = E~X / E: Dt + C\E~X We Obtain
Jo
/ = E- 2xe 2x* F *22 J -R.2 2 Rx +2 X
*2er2 =
r
Y X 2xe
/ E1 Dt — X E Dt —
2c\Xe 2cixe Jo
Jo
Substituting Into The Differential Equation, We
Have
