, Handbook of Exact
Solutions to
Mathematical Equations
Andrei D. Polyanin
,First edition published 2025
by CRC Press
2385 Executive Center Drive, Suite 320, Boca Raton, FL 33431
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2025 Andrei D. Polyanin
Reasonable efforts have been made to publish reliable data and information, but the author and publisher
cannot as- sume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication and
apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright
material has not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter
invented, including pho- tocopying, microfilming, and recording, or in any information storage or retrieval
system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.com or
contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For
works that are not available on CCC please contact
Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for
iden- tification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Names: Poĺiànin, A. D. (Andreĭ Dmitrievich), author.
Title: Handbook of exact solutions to mathematical equations / A.D.
Polyanin.
Description: First edition. | Boca Raton, FL : CRC Press, 2025. | Series:
Advances in applied mathematics | Includes bibliographical
references and index.
Identifiers: LCCN 2024005775 | ISBN 9780367507992 (hardback) | ISBN
9781032807232 (paperback) | ISBN 9781003051329 (ebook)
Subjects: LCSH: Differential equations, Partial--Numerical
solutions--Handbooks, manuals, etc. | Differential
equations, Nonlinear--Numerical solutions--Handbooks,
manuals, etc.
Classification: LCC QA377 .P5678 2025 | DDC 515/.35--dc23/eng/20240412 LC
record available at https://lccn.loc.gov/2024005775
ISBN: 978-0-367-50799-2 (hbk)
ISBN: 978-1-032-80723-2 (pbk)
ISBN: 978-1-003-05132-9 (ebk) DOI:
10.1201/9781003051329
Typeset in CMR10 font
by KnowledgeWorks Global Ltd.
,Contents
Preface xi
Author xv
Notations and Remarks xvii
1 Algebraic and Transcendental Equations 1
1.1. Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1. Linear and Quadratic Equations . . . . . . . . . . . . . . . . . . 1
1.1.2. Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3. Equations of the Fourth Degree . . . . . . . . . . . . . . . . . . 5
1.1.4. Equations of the Fifth Degree . . . . . . . . . . . . . . . . . . . 9
1.1.5. Algebraic Equations of Arbitrary Degree . . . . . . . . . . . . . 12
1.1.6. Systems of Linear Algebraic Equations . . . . . . . . . . . . . . 19
1.2. Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.1. Binomial Trigonometric Equations . . . . . . . . . . . . . . . . . 20
1.2.2. Trigonometric Equations Containing Several Terms . . . . . . . . 21
1.2.3. Trigonometric Equations of the General Form . . . . . . . . . . . 27
1.3. Other Transcendental Equations . . . . . . . . . . . . . . . . . . . . . . 29
1.3.1. Equations Containing Exponential Functions . . . . . . . . . . . 29
1.3.2. Equations Containing Hyperbolic Functions . . . . . . . . . . . . 30
1.3.3. Equations Containing Logarithmic Functions . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Ordinary Differential Equations 37
2.1. First-Order Ordinary Differential Equations . . . . . . . . . . . . . . . . 37
2.1.1. Simplest First-Order ODEs . . . . . . . . . . . . . . . . . . . . . 37
2.1.2. Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.3. Abel Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.4. Other First-Order ODEs Solved for the Derivative . . . . . . . . . 46
2.1.5. ODEs Not Solved for the Derivative and ODEs Defined
Parametrically . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2. Second-Order Linear Ordinary Differential Equations . . . . . . . . . . 50
2.2.1. Preliminary Remarks and Some Formulas . . . . . . . . . . . . . 50
2.2.2. Equations Involving Power Functions . . . . . . . . . . . . . . . 51
2.2.3. Equations Involving Exponential and Other Elementary Functions 69
,vi C ONTENTS
2.2.4. Equations Involving Arbitrary Functions ................................................72
2.3. Second-Order Nonlinear Ordinary Differential Equations ............................... 74
2.3.1. Equations of the Form yxx′′ = f(x, y) .................................................... 74
2.3.2. Equations of the Form f(x,y)y ′′xx = g(x, y, y′ )x....................................... 77
2.3.3. ODEs of General Form Containing Arbitrary Functions of Two
Arguments ................................................................................................ 82
2.4. Higher-Order Ordinary Differential Equations ................................................... 86
2.4.1. Higher-Order Linear Ordinary Differential Equations .......................... 86
2.4.2. Third- and Fourth-Order Nonlinear Ordinary Differential Equations 100
2.4.3. Higher-Order Nonlinear Ordinary Differential Equations ................... 102
References ..................................................................................................................... 109
3 Systems of Ordinary Differential Equations 111
3.1. Linear Systems of ODEs........................................................................................ 111
3.1.1. Systems of Two First-Order ODEs .......................................................... 111
3.1.2. Systems of Two Second-Order ODEs.................................................... 114
3.1.3. Other Systems of Two ODEs.................................................................. 120
3.1.4. Systems of Three and More ODEs ......................................................... 121
3.2. Nonlinear Systems of Two ODEs ........................................................................ 123
3.2.1. Systems of First-Order ODEs ................................................................. 123
3.2.2. Systems of Second- and Third-Order ODEs ......................................... 133
3.3. Nonlinear Systems of Three or More ODEs ...................................................... 141
3.3.1. Systems of Three ODEs .......................................................................... 141
3.3.2. Equations of Dynamics of a Rigid Body with a Fixed Point................ 144
References ..................................................................................................................... 147
4.1.1. Preliminary Remarks. Solution Methods ............................................. 149
4 First-Order
4.1.2. Partial
Equations Differential
of the Form Equations
f(x, y)ux + g(x, y)uy = 0 .....................................149 150
4.1.3. Equations of the Form f(x, y)u
4.1. Linear Partial Differential Equations in xTwo Independent + g(x, y)u y = h(x, y) ..........................
Variables . . . 149 153
4.1.4. Equations of the Form f(x, y)ux + g(x, y)uy = h(x, y)u + r(x, y) 155
4.2. Quasilinear Partial Differential Equations in Two Independent Variables . 158
4.2.1. Preliminary Remarks. Solution Methods ............................................. 158
4.2.2. Equations of the Form f(x, y)ux + g(x, y)uy = h(x, y, u) .......................... 159
4.2.3. Equations of the Form ux + f(x, y, u)uy = 0 ............................................ 161
4.2.4. Equations of the Form ux + f(x, y, u)uy = g(x, y, u) ................................. 164
4.3. Nonlinear Partial Differential Equations in Two Independent Variables . . 167
4.3.1. Preliminary Remarks. A Complete Integral.......................................... 167
4.3.2. Equations Quadratic in One Derivative ..................................................... 168
4.3.3. Equations Quadratic in Two Derivatives................................................ 171
4.3.4. Equations with Arbitrary Nonlinearities in Derivatives....................... 173
References ..................................................................................................................... 178
, C ONTENTS vii
5 Linear Equations and Problems of Mathematical Physics 179
5.1. Parabolic Equations ............................................................................................. 179
5.1.1. Heat (Diffusion) Equation ut = auxx ....................................................... 179
5.1.2. Nonhomogeneous Heat Equation ut = auxx + Φ(x, t) ........................... 181
5.1.3. Heat Type Equation of the Form ut = auxx + bux + cu + Φ(x, t) 183
5.1.4. Heat Equation with Axial Symmetry ut = a(urr + r−1ur) ........................ 183
5.1.5. Nonhomogeneous Heat Equation with Axial Symmetry
ut = a(urr + r−1u r) + Φ(r, t) ........................................................................185
5.1.6. Heat Equation with Central Symmetry ut = a(urr + 2r−1ur) . . 186
5.1.7. Nonhomogeneous Heat Equation with Central Symmetry
ut = a(urr + 2r−1ur) + Φ(r, t) ..................................................................... 188
5.1.8. Heat Type Equation of the Form ut = uxx + (1 − 2β)x−1u x . . . 189
5.1.9. Heat Type Equation of the Form ut = [f(x)ux]x ..................................... 190
5.1.10. Equations of the Form s(x)ut = [p(x)ux ]x − q(x)u + Φ(x,t) . . 191
2
5.1.11. Liquid-Film Mass Transfer Equation (1 − y )ux = auyy .........................193
5.1.12. Equations of the Diffusion (Thermal) Boundary Layer....................... 195
ℏ2
5.1.13. Schrödinger Equation iℏut = − 2m uxx + U(x)u ..................................... 196
5.2. Hyperbolic Equations .......................................................................................... 198
5.2.1. Wave Equation utt = a2uxx ....................................................................... 198
5.2.2. Nonhomogeneous Wave Equation utt = a2uxx + Φ(x, t)........................ 199
5.2.3. Klein–Gordon Equation utt = a2uxx − bu ............................................. 200
5.2.4. Nonhomogeneous Klein–Gordon Equation
utt = a2uxx − bu + Φ(x, t) ...........................................................................201
5.2.5. Wave Equation with Axial Symmetry
utt = a2(urr + r−1ur) + Φ(r, t) ..................................................................... 202
5.2.6. Wave Equation with Central Symmetry
utt = a2(urr + 2r−1ur) + Φ(r, t) ................................................................... 204
5.2.7. Equations of the Form s(x)utt = [p(x)ux ]x − q(x)u + Φ(x,t) . 205
2
5.2.8. Telegraph Type Equations utt + kut = a uxx + bux + cu + Φ(x, t) 206
5.3. Elliptic Equations ................................................................................................. 207
5.3.1. Laplace Equation ∆u = 0 ....................................................................... 207
5.3.2. Poisson Equation ∆u + Φ(x, y) = 0 ........................................................210
5.3.3. Helmholtz Equation ∆u + λu = −Φ(x, y) ............................................. 212
5.3.4. Convective Heat and Mass Transfer Equations ................................... 216
5.3.5. Equations of Heat and Mass Transfer in Anisotropic Media .............. 221
5.3.6. Tricomi and Related Equations .............................................................228
5.4. Simplifications of Second-Order Linear Partial Differential Equations . . 231
5.4.1. Reduction of PDEs in Two Independent Variables to Canonical
Forms........................................................................................................ 231
5.4.2. Simplifications of Linear Constant-Coefficient Partial Differential
Equations ................................................................................................. 233
5.5. Third-Order Linear Partial Differential Equations ............................................. 235
5.5.1. Equations Containing the First Derivative in t and the Third
Derivative in x .......................................................................................... 235
5.5.2. Equations Containing the First Derivative in t and a Mixed Third
Derivative........................................................................................................... 236
,viii C ONTENTS
5.5.3. Equations Containing the Second Derivative in t and a Mixed
Third Derivative ................................................................................................ 242
5.6. Fourth-Order Linear Partial Differential Equations ......................................... 244
5.6.1. Equation of Transverse Vibration of an Elastic Rod
utt + a2uxxxx = 0 .............................................................................. 244
5.6.2. Nonhomogeneous Equation of the Form utt + a2uxxxx = Φ(x, t) 245
5.6.3. Biharmonic Equation ∆∆u = 0 ............................................................ 247
5.6.4. Nonhomogeneous Biharmonic Equation ∆∆u = Φ(x, y) ...................248
References .................................................................................................................... 249
6 Nonlinear Equations of Mathematical Physics 251
6.1. Parabolic Equations ............................................................................................. 252
6.1.1. Quasilinear Heat Equations with a Source of the Form
ut = auxx + f(u) ................................................................................ 252
6.1.2. Burgers Type Equations and Related PDEs ......................................... 256
6.1.3. Reaction-Diffusion Equations of the Form ut = [f(u)ux]x + g(u) 260
6.1.4. Other Reaction-Diffusion and Heat PDEs with Variable Transfer
Coefficient............................................................................................... 268
6.1.5. Convection-Diffusion Type PDEs ........................................................... 272
6.1.6. Nonlinear Schrödinger Equations and Related PDEs ......................... 277
6.2. Hyperbolic Equations ..........................................................................................282
6.2.1. Nonlinear Klein–Gordon Equations of the Form utt = auxx + f(u) 282
6.2.2. Other Nonlinear Wave Type Equations ................................................ 287
6.3. Elliptic Equations ................................................................................................. 295
6.3.1. Heat Equations with Nonlinear Source of the Form
uxx + uyy = f(u) ......................................................................................... 295
6.3.2. Stationary Anisotropic Heat/Diffusion Equations of the Form
[f(x)ux]x + [g(y)uy]y = h(u) ..................................................................... 298
6.3.3. Stationary Anisotropic Heat/Diffusion Equations of the Form
[f(u)ux]x + [g(u)uy]y = h(u) ..................................................................... 299
6.4. Other Second-Order Equations ......................................................................... 302
6.4.1. Equations of Transonic Gas Flow ......................................................... 302
6.4.2. Monge–Ampère Type Equations .......................................................... 303
6.5. Higher-Order Equations...................................................................................... 306
6.5.1. Third-Order Equations ........................................................................... 306
6.5.2. Fourth-Order Equations ........................................................................ 324
References .................................................................................................................... 329
7 Systems of Partial Differential Equations 335
7.1. Systems of Two First-Order PDEs .......................................................................335
7.1.1. Linear Systems of Two First-Order PDEs ..............................................335
7.1.2. Nonlinear Systems of the Form ux = F(u, w), wt = G(u, w) . . 336
7.1.3. Gas Dynamic Type Systems Linearizable with the Hodograph
Transformation........................................................................................ 341
7.2. Systems of Two Second-Order PDEs................................................................. 347
7.2.1. Linear Systems of Two Second-Order PDEs........................................ 347
, C ONTENTS ix
7.2.2. Nonlinear Parabolic Systems of the Form
ut = auxx + F (u, w), wt = bwxx + G(u, w) ................................................ 349
7.2.3. Nonlinear Parabolic Systems of the Form
ut = ax−n(xnux) x + F(u, w), wt = bx−n(xnwx) x + G(u, w) . 363
7.2.4. Nonlinear Hyperbolic Systems of the Form
utt = auxx + F (u, w), wtt = bwxx + G(u, w) ............................................ 369
7.2.5. Nonlinear Hyperbolic Systems of the Form
utt = ax−n(xnux) x + F(u, w), wtt = bx−n(xnwx) x + G(u, w) 376
7.2.6. Nonlinear Elliptic Systems of the Form
∆u = F (u, w), ∆w = G(u, w) ................................................................... 380
7.3. PDE Systems of General Form............................................................................ 384
7.3.1. Linear Systems ....................................................................................... 384
7.3.2. Nonlinear Systems of Two Equations Involving the First
Derivatives with Respect to t ............................................................... 385
7.3.3. Nonlinear Systems of Two Equations Involving the Second
Derivatives with Respect to t
389
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
8 Integral Equations 395
8.1. Integral Equations of the First Kind with Variable Limit of Integration . . 395
8.1.1. Linear Volterra Integral Equations of the First Kind........................... 395
8.1.2. Nonlinear Volterra Integral Equations of the First Kind .................... 403
8.2. Integral Equations of the Second Kind with Variable Limit of Integration 406
8.2.1. Linear Volterra Integral Equations of the Second Kind ..................... 406
8.2.2. Nonlinear Volterra Integral Equations of the Second Kind ............... 424
8.3. Equations of the First Kind with Constant Limits of Integration ................... 428
8.3.1. Linear Fredholm Integral Equations of the First Kind ....................... 428
8.3.2. Nonlinear Fredholm Integral Equations of the First Kind ................. 437
8.4. Equations of the Second Kind with Constant Limits of Integration.............. 439
8.4.1. Linear Fredholm Integral Equations of the Second Kind .................. 439
8.4.2. Nonlinear Fredholm Integral Equations of the Second Kind ............ 450
References .................................................................................................................... 455
9 Difference and Functional Equations 457
9.1. Difference Equations .......................................................................................... 457
9.1.1. Difference Equations with Discrete Argument .................................. 457
9.1.2. Difference Equations with Continuous Argument ............................. 465
9.2. Linear Functional Equations in One Independent Variable ............................ 480
9.2.1. Linear Functional Equations Involving Unknown Function with
Two Different Arguments ..................................................................... 480
9.2.2. Other Linear Functional Equations ...................................................... 488
9.3. Nonlinear Functional Equations in One Independent Variable...................... 492
9.3.1. Functional Equations with Quadratic Nonlinearity............................ 492
9.3.2. Functional Equations with Power Nonlinearity.................................. 496
9.3.3. Nonlinear Functional Equation of General Form ............................... 497
9.4. Functional Equations in Several Independent Variables.................................. 501
9.4.1. Linear Functional Equations .................................................................. 501
,x C ONTENTS
9.4.2. Nonlinear Functional Equations . . . . . . . . . . . . . . . . . .
507
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
10 Ordinary Functional Differential Equations 519
10.1. First-Order Linear Ordinary Functional Differential Equations ....................... 519
10.1.1. ODEs with Constant Delays ................................................................... 519
10.1.2. Pantograph-Type ODEs with Proportional Arguments ...................... 522
10.1.3. Other Ordinary Functional Differential Equations ............................. 524
10.2. First-Order Nonlinear Ordinary Functional Differential Equations................. 525
10.2.1. ODEs with Constant Delays ................................................................... 525
10.2.2. Pantograph-Type ODEs with Proportional Arguments ...................... 527
10.2.3. Other Ordinary Functional Differential Equations ............................. 530
10.3. Second-Order Linear Ordinary Functional Differential Equations.................. 531
10.3.1. ODEs with Constant Delays ................................................................... 531
10.3.2. Pantograph-Type ODEs with Proportional Arguments ......................533
10.3.3. Other Ordinary Functional Differential Equations ..............................535
10.4. Second-Order Nonlinear Ordinary Functional Differential Equations . . . 536
10.4.1. ODEs with Constant Delays .................................................................. 536
10.4.2. Pantograph-Type ODEs with Proportional Arguments ...................... 537
10.4.3. Other Ordinary Functional Differential Equations ............................. 538
10.5. Higher-Order Ordinary Functional Differential Equations .............................. 539
10.5.1. Linear Ordinary Functional Differential Equations ............................ 539
10.5.2. Nonlinear Ordinary Functional Differential Equations ...................... 542
References .................................................................................................................... 543
11 Partial Functional Differential Equations 545
11.1. Linear Partial Functional Differential Equations .............................................. 546
11.1.1. PDEs with Constant Delay .................................................................... 546
11.1.2. PDEs with Proportional Delay .............................................................. 554
11.1.3. PDEs with Anisotropic Time Delay ....................................................... 560
11.2. Nonlinear PDEs with Constant Delays .............................................................. 562
11.2.1. Parabolic Equations ............................................................................... 562
11.2.2. Hyperbolic Equations ............................................................................. 581
11.3. Nonlinear PDEs with Proportional Arguments ................................................ 590
11.3.1. Parabolic Equations ............................................................................... 590
11.3.2. Hyperbolic Equations ............................................................................ 600
11.4. Partial Functional Differential Equations with Arguments of General Type 605
11.4.1. Parabolic Equations ............................................................................... 605
11.4.2. Hyperbolic Equations ............................................................................. 613
11.5. PDEs with Anisotropic Time Delay ..................................................................... 617
11.5.1. Parabolic Equations ................................................................................ 617
11.5.2. Hyperbolic Equations ............................................................................. 619
References ..................................................................................................................... 621
Index 623
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Solutions to
Mathematical Equations
Andrei D. Polyanin
,First edition published 2025
by CRC Press
2385 Executive Center Drive, Suite 320, Boca Raton, FL 33431
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2025 Andrei D. Polyanin
Reasonable efforts have been made to publish reliable data and information, but the author and publisher
cannot as- sume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication and
apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright
material has not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter
invented, including pho- tocopying, microfilming, and recording, or in any information storage or retrieval
system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.com or
contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For
works that are not available on CCC please contact
Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for
iden- tification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Names: Poĺiànin, A. D. (Andreĭ Dmitrievich), author.
Title: Handbook of exact solutions to mathematical equations / A.D.
Polyanin.
Description: First edition. | Boca Raton, FL : CRC Press, 2025. | Series:
Advances in applied mathematics | Includes bibliographical
references and index.
Identifiers: LCCN 2024005775 | ISBN 9780367507992 (hardback) | ISBN
9781032807232 (paperback) | ISBN 9781003051329 (ebook)
Subjects: LCSH: Differential equations, Partial--Numerical
solutions--Handbooks, manuals, etc. | Differential
equations, Nonlinear--Numerical solutions--Handbooks,
manuals, etc.
Classification: LCC QA377 .P5678 2025 | DDC 515/.35--dc23/eng/20240412 LC
record available at https://lccn.loc.gov/2024005775
ISBN: 978-0-367-50799-2 (hbk)
ISBN: 978-1-032-80723-2 (pbk)
ISBN: 978-1-003-05132-9 (ebk) DOI:
10.1201/9781003051329
Typeset in CMR10 font
by KnowledgeWorks Global Ltd.
,Contents
Preface xi
Author xv
Notations and Remarks xvii
1 Algebraic and Transcendental Equations 1
1.1. Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1. Linear and Quadratic Equations . . . . . . . . . . . . . . . . . . 1
1.1.2. Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3. Equations of the Fourth Degree . . . . . . . . . . . . . . . . . . 5
1.1.4. Equations of the Fifth Degree . . . . . . . . . . . . . . . . . . . 9
1.1.5. Algebraic Equations of Arbitrary Degree . . . . . . . . . . . . . 12
1.1.6. Systems of Linear Algebraic Equations . . . . . . . . . . . . . . 19
1.2. Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.1. Binomial Trigonometric Equations . . . . . . . . . . . . . . . . . 20
1.2.2. Trigonometric Equations Containing Several Terms . . . . . . . . 21
1.2.3. Trigonometric Equations of the General Form . . . . . . . . . . . 27
1.3. Other Transcendental Equations . . . . . . . . . . . . . . . . . . . . . . 29
1.3.1. Equations Containing Exponential Functions . . . . . . . . . . . 29
1.3.2. Equations Containing Hyperbolic Functions . . . . . . . . . . . . 30
1.3.3. Equations Containing Logarithmic Functions . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Ordinary Differential Equations 37
2.1. First-Order Ordinary Differential Equations . . . . . . . . . . . . . . . . 37
2.1.1. Simplest First-Order ODEs . . . . . . . . . . . . . . . . . . . . . 37
2.1.2. Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.3. Abel Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.4. Other First-Order ODEs Solved for the Derivative . . . . . . . . . 46
2.1.5. ODEs Not Solved for the Derivative and ODEs Defined
Parametrically . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2. Second-Order Linear Ordinary Differential Equations . . . . . . . . . . 50
2.2.1. Preliminary Remarks and Some Formulas . . . . . . . . . . . . . 50
2.2.2. Equations Involving Power Functions . . . . . . . . . . . . . . . 51
2.2.3. Equations Involving Exponential and Other Elementary Functions 69
,vi C ONTENTS
2.2.4. Equations Involving Arbitrary Functions ................................................72
2.3. Second-Order Nonlinear Ordinary Differential Equations ............................... 74
2.3.1. Equations of the Form yxx′′ = f(x, y) .................................................... 74
2.3.2. Equations of the Form f(x,y)y ′′xx = g(x, y, y′ )x....................................... 77
2.3.3. ODEs of General Form Containing Arbitrary Functions of Two
Arguments ................................................................................................ 82
2.4. Higher-Order Ordinary Differential Equations ................................................... 86
2.4.1. Higher-Order Linear Ordinary Differential Equations .......................... 86
2.4.2. Third- and Fourth-Order Nonlinear Ordinary Differential Equations 100
2.4.3. Higher-Order Nonlinear Ordinary Differential Equations ................... 102
References ..................................................................................................................... 109
3 Systems of Ordinary Differential Equations 111
3.1. Linear Systems of ODEs........................................................................................ 111
3.1.1. Systems of Two First-Order ODEs .......................................................... 111
3.1.2. Systems of Two Second-Order ODEs.................................................... 114
3.1.3. Other Systems of Two ODEs.................................................................. 120
3.1.4. Systems of Three and More ODEs ......................................................... 121
3.2. Nonlinear Systems of Two ODEs ........................................................................ 123
3.2.1. Systems of First-Order ODEs ................................................................. 123
3.2.2. Systems of Second- and Third-Order ODEs ......................................... 133
3.3. Nonlinear Systems of Three or More ODEs ...................................................... 141
3.3.1. Systems of Three ODEs .......................................................................... 141
3.3.2. Equations of Dynamics of a Rigid Body with a Fixed Point................ 144
References ..................................................................................................................... 147
4.1.1. Preliminary Remarks. Solution Methods ............................................. 149
4 First-Order
4.1.2. Partial
Equations Differential
of the Form Equations
f(x, y)ux + g(x, y)uy = 0 .....................................149 150
4.1.3. Equations of the Form f(x, y)u
4.1. Linear Partial Differential Equations in xTwo Independent + g(x, y)u y = h(x, y) ..........................
Variables . . . 149 153
4.1.4. Equations of the Form f(x, y)ux + g(x, y)uy = h(x, y)u + r(x, y) 155
4.2. Quasilinear Partial Differential Equations in Two Independent Variables . 158
4.2.1. Preliminary Remarks. Solution Methods ............................................. 158
4.2.2. Equations of the Form f(x, y)ux + g(x, y)uy = h(x, y, u) .......................... 159
4.2.3. Equations of the Form ux + f(x, y, u)uy = 0 ............................................ 161
4.2.4. Equations of the Form ux + f(x, y, u)uy = g(x, y, u) ................................. 164
4.3. Nonlinear Partial Differential Equations in Two Independent Variables . . 167
4.3.1. Preliminary Remarks. A Complete Integral.......................................... 167
4.3.2. Equations Quadratic in One Derivative ..................................................... 168
4.3.3. Equations Quadratic in Two Derivatives................................................ 171
4.3.4. Equations with Arbitrary Nonlinearities in Derivatives....................... 173
References ..................................................................................................................... 178
, C ONTENTS vii
5 Linear Equations and Problems of Mathematical Physics 179
5.1. Parabolic Equations ............................................................................................. 179
5.1.1. Heat (Diffusion) Equation ut = auxx ....................................................... 179
5.1.2. Nonhomogeneous Heat Equation ut = auxx + Φ(x, t) ........................... 181
5.1.3. Heat Type Equation of the Form ut = auxx + bux + cu + Φ(x, t) 183
5.1.4. Heat Equation with Axial Symmetry ut = a(urr + r−1ur) ........................ 183
5.1.5. Nonhomogeneous Heat Equation with Axial Symmetry
ut = a(urr + r−1u r) + Φ(r, t) ........................................................................185
5.1.6. Heat Equation with Central Symmetry ut = a(urr + 2r−1ur) . . 186
5.1.7. Nonhomogeneous Heat Equation with Central Symmetry
ut = a(urr + 2r−1ur) + Φ(r, t) ..................................................................... 188
5.1.8. Heat Type Equation of the Form ut = uxx + (1 − 2β)x−1u x . . . 189
5.1.9. Heat Type Equation of the Form ut = [f(x)ux]x ..................................... 190
5.1.10. Equations of the Form s(x)ut = [p(x)ux ]x − q(x)u + Φ(x,t) . . 191
2
5.1.11. Liquid-Film Mass Transfer Equation (1 − y )ux = auyy .........................193
5.1.12. Equations of the Diffusion (Thermal) Boundary Layer....................... 195
ℏ2
5.1.13. Schrödinger Equation iℏut = − 2m uxx + U(x)u ..................................... 196
5.2. Hyperbolic Equations .......................................................................................... 198
5.2.1. Wave Equation utt = a2uxx ....................................................................... 198
5.2.2. Nonhomogeneous Wave Equation utt = a2uxx + Φ(x, t)........................ 199
5.2.3. Klein–Gordon Equation utt = a2uxx − bu ............................................. 200
5.2.4. Nonhomogeneous Klein–Gordon Equation
utt = a2uxx − bu + Φ(x, t) ...........................................................................201
5.2.5. Wave Equation with Axial Symmetry
utt = a2(urr + r−1ur) + Φ(r, t) ..................................................................... 202
5.2.6. Wave Equation with Central Symmetry
utt = a2(urr + 2r−1ur) + Φ(r, t) ................................................................... 204
5.2.7. Equations of the Form s(x)utt = [p(x)ux ]x − q(x)u + Φ(x,t) . 205
2
5.2.8. Telegraph Type Equations utt + kut = a uxx + bux + cu + Φ(x, t) 206
5.3. Elliptic Equations ................................................................................................. 207
5.3.1. Laplace Equation ∆u = 0 ....................................................................... 207
5.3.2. Poisson Equation ∆u + Φ(x, y) = 0 ........................................................210
5.3.3. Helmholtz Equation ∆u + λu = −Φ(x, y) ............................................. 212
5.3.4. Convective Heat and Mass Transfer Equations ................................... 216
5.3.5. Equations of Heat and Mass Transfer in Anisotropic Media .............. 221
5.3.6. Tricomi and Related Equations .............................................................228
5.4. Simplifications of Second-Order Linear Partial Differential Equations . . 231
5.4.1. Reduction of PDEs in Two Independent Variables to Canonical
Forms........................................................................................................ 231
5.4.2. Simplifications of Linear Constant-Coefficient Partial Differential
Equations ................................................................................................. 233
5.5. Third-Order Linear Partial Differential Equations ............................................. 235
5.5.1. Equations Containing the First Derivative in t and the Third
Derivative in x .......................................................................................... 235
5.5.2. Equations Containing the First Derivative in t and a Mixed Third
Derivative........................................................................................................... 236
,viii C ONTENTS
5.5.3. Equations Containing the Second Derivative in t and a Mixed
Third Derivative ................................................................................................ 242
5.6. Fourth-Order Linear Partial Differential Equations ......................................... 244
5.6.1. Equation of Transverse Vibration of an Elastic Rod
utt + a2uxxxx = 0 .............................................................................. 244
5.6.2. Nonhomogeneous Equation of the Form utt + a2uxxxx = Φ(x, t) 245
5.6.3. Biharmonic Equation ∆∆u = 0 ............................................................ 247
5.6.4. Nonhomogeneous Biharmonic Equation ∆∆u = Φ(x, y) ...................248
References .................................................................................................................... 249
6 Nonlinear Equations of Mathematical Physics 251
6.1. Parabolic Equations ............................................................................................. 252
6.1.1. Quasilinear Heat Equations with a Source of the Form
ut = auxx + f(u) ................................................................................ 252
6.1.2. Burgers Type Equations and Related PDEs ......................................... 256
6.1.3. Reaction-Diffusion Equations of the Form ut = [f(u)ux]x + g(u) 260
6.1.4. Other Reaction-Diffusion and Heat PDEs with Variable Transfer
Coefficient............................................................................................... 268
6.1.5. Convection-Diffusion Type PDEs ........................................................... 272
6.1.6. Nonlinear Schrödinger Equations and Related PDEs ......................... 277
6.2. Hyperbolic Equations ..........................................................................................282
6.2.1. Nonlinear Klein–Gordon Equations of the Form utt = auxx + f(u) 282
6.2.2. Other Nonlinear Wave Type Equations ................................................ 287
6.3. Elliptic Equations ................................................................................................. 295
6.3.1. Heat Equations with Nonlinear Source of the Form
uxx + uyy = f(u) ......................................................................................... 295
6.3.2. Stationary Anisotropic Heat/Diffusion Equations of the Form
[f(x)ux]x + [g(y)uy]y = h(u) ..................................................................... 298
6.3.3. Stationary Anisotropic Heat/Diffusion Equations of the Form
[f(u)ux]x + [g(u)uy]y = h(u) ..................................................................... 299
6.4. Other Second-Order Equations ......................................................................... 302
6.4.1. Equations of Transonic Gas Flow ......................................................... 302
6.4.2. Monge–Ampère Type Equations .......................................................... 303
6.5. Higher-Order Equations...................................................................................... 306
6.5.1. Third-Order Equations ........................................................................... 306
6.5.2. Fourth-Order Equations ........................................................................ 324
References .................................................................................................................... 329
7 Systems of Partial Differential Equations 335
7.1. Systems of Two First-Order PDEs .......................................................................335
7.1.1. Linear Systems of Two First-Order PDEs ..............................................335
7.1.2. Nonlinear Systems of the Form ux = F(u, w), wt = G(u, w) . . 336
7.1.3. Gas Dynamic Type Systems Linearizable with the Hodograph
Transformation........................................................................................ 341
7.2. Systems of Two Second-Order PDEs................................................................. 347
7.2.1. Linear Systems of Two Second-Order PDEs........................................ 347
, C ONTENTS ix
7.2.2. Nonlinear Parabolic Systems of the Form
ut = auxx + F (u, w), wt = bwxx + G(u, w) ................................................ 349
7.2.3. Nonlinear Parabolic Systems of the Form
ut = ax−n(xnux) x + F(u, w), wt = bx−n(xnwx) x + G(u, w) . 363
7.2.4. Nonlinear Hyperbolic Systems of the Form
utt = auxx + F (u, w), wtt = bwxx + G(u, w) ............................................ 369
7.2.5. Nonlinear Hyperbolic Systems of the Form
utt = ax−n(xnux) x + F(u, w), wtt = bx−n(xnwx) x + G(u, w) 376
7.2.6. Nonlinear Elliptic Systems of the Form
∆u = F (u, w), ∆w = G(u, w) ................................................................... 380
7.3. PDE Systems of General Form............................................................................ 384
7.3.1. Linear Systems ....................................................................................... 384
7.3.2. Nonlinear Systems of Two Equations Involving the First
Derivatives with Respect to t ............................................................... 385
7.3.3. Nonlinear Systems of Two Equations Involving the Second
Derivatives with Respect to t
389
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
8 Integral Equations 395
8.1. Integral Equations of the First Kind with Variable Limit of Integration . . 395
8.1.1. Linear Volterra Integral Equations of the First Kind........................... 395
8.1.2. Nonlinear Volterra Integral Equations of the First Kind .................... 403
8.2. Integral Equations of the Second Kind with Variable Limit of Integration 406
8.2.1. Linear Volterra Integral Equations of the Second Kind ..................... 406
8.2.2. Nonlinear Volterra Integral Equations of the Second Kind ............... 424
8.3. Equations of the First Kind with Constant Limits of Integration ................... 428
8.3.1. Linear Fredholm Integral Equations of the First Kind ....................... 428
8.3.2. Nonlinear Fredholm Integral Equations of the First Kind ................. 437
8.4. Equations of the Second Kind with Constant Limits of Integration.............. 439
8.4.1. Linear Fredholm Integral Equations of the Second Kind .................. 439
8.4.2. Nonlinear Fredholm Integral Equations of the Second Kind ............ 450
References .................................................................................................................... 455
9 Difference and Functional Equations 457
9.1. Difference Equations .......................................................................................... 457
9.1.1. Difference Equations with Discrete Argument .................................. 457
9.1.2. Difference Equations with Continuous Argument ............................. 465
9.2. Linear Functional Equations in One Independent Variable ............................ 480
9.2.1. Linear Functional Equations Involving Unknown Function with
Two Different Arguments ..................................................................... 480
9.2.2. Other Linear Functional Equations ...................................................... 488
9.3. Nonlinear Functional Equations in One Independent Variable...................... 492
9.3.1. Functional Equations with Quadratic Nonlinearity............................ 492
9.3.2. Functional Equations with Power Nonlinearity.................................. 496
9.3.3. Nonlinear Functional Equation of General Form ............................... 497
9.4. Functional Equations in Several Independent Variables.................................. 501
9.4.1. Linear Functional Equations .................................................................. 501
,x C ONTENTS
9.4.2. Nonlinear Functional Equations . . . . . . . . . . . . . . . . . .
507
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
10 Ordinary Functional Differential Equations 519
10.1. First-Order Linear Ordinary Functional Differential Equations ....................... 519
10.1.1. ODEs with Constant Delays ................................................................... 519
10.1.2. Pantograph-Type ODEs with Proportional Arguments ...................... 522
10.1.3. Other Ordinary Functional Differential Equations ............................. 524
10.2. First-Order Nonlinear Ordinary Functional Differential Equations................. 525
10.2.1. ODEs with Constant Delays ................................................................... 525
10.2.2. Pantograph-Type ODEs with Proportional Arguments ...................... 527
10.2.3. Other Ordinary Functional Differential Equations ............................. 530
10.3. Second-Order Linear Ordinary Functional Differential Equations.................. 531
10.3.1. ODEs with Constant Delays ................................................................... 531
10.3.2. Pantograph-Type ODEs with Proportional Arguments ......................533
10.3.3. Other Ordinary Functional Differential Equations ..............................535
10.4. Second-Order Nonlinear Ordinary Functional Differential Equations . . . 536
10.4.1. ODEs with Constant Delays .................................................................. 536
10.4.2. Pantograph-Type ODEs with Proportional Arguments ...................... 537
10.4.3. Other Ordinary Functional Differential Equations ............................. 538
10.5. Higher-Order Ordinary Functional Differential Equations .............................. 539
10.5.1. Linear Ordinary Functional Differential Equations ............................ 539
10.5.2. Nonlinear Ordinary Functional Differential Equations ...................... 542
References .................................................................................................................... 543
11 Partial Functional Differential Equations 545
11.1. Linear Partial Functional Differential Equations .............................................. 546
11.1.1. PDEs with Constant Delay .................................................................... 546
11.1.2. PDEs with Proportional Delay .............................................................. 554
11.1.3. PDEs with Anisotropic Time Delay ....................................................... 560
11.2. Nonlinear PDEs with Constant Delays .............................................................. 562
11.2.1. Parabolic Equations ............................................................................... 562
11.2.2. Hyperbolic Equations ............................................................................. 581
11.3. Nonlinear PDEs with Proportional Arguments ................................................ 590
11.3.1. Parabolic Equations ............................................................................... 590
11.3.2. Hyperbolic Equations ............................................................................ 600
11.4. Partial Functional Differential Equations with Arguments of General Type 605
11.4.1. Parabolic Equations ............................................................................... 605
11.4.2. Hyperbolic Equations ............................................................................. 613
11.5. PDEs with Anisotropic Time Delay ..................................................................... 617
11.5.1. Parabolic Equations ................................................................................ 617
11.5.2. Hyperbolic Equations ............................................................................. 619
References ..................................................................................................................... 621
Index 623
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