, Handbook of Exact
Solutions to Mathematical
Equations
This reference book describes the exact solutions of the following types of mathematical equa-
tions:
• Algebraic and Transcendental Equations
• Ordinary Differential Equations
• Systems of Ordinary Differential Equations
• First-Order Partial Differential Equations
• Linear Equations and Problems of Mathematical Physics
• Nonlinear Equations of Mathematical Physics
• Systems of Partial Differential Equations
• Integral Equations
• Difference and Functional Equations
• Ordinary Functional Differential Equations
• Partial Functional Differential Equations
The book pays special attention to equations found in various fields of natural and engineering
sciences (in the theory of heat and mass transfer, wave theory, hydrodynamics, gas dynamics,
combustion theory, elasticity theory, general mechanics, theoretical physics, nonlinear optics,
biology, chemical engineering sciences, ecology, and others) and equations of a reasonably gen-
eral form, which depend on free parameters or arbitrary functions.
The Handbook of Exact Solutions to Mathematical Equations is unique in world literature and
contains extensive new material. The exact solutions given in the book, being rigorous math-
ematical standards, can be used as test problems to assess the accuracy and verify the adequacy
of various numerical and approximate analytical methods for solving mathematical equations, as
well as for checking and comparing the effectiveness of exact analytical methods.
,Advances in Applied Mathematics
Series Editor: Daniel Zwillinger
Boundary Value Problems on Time Scales, Volume I
Svetlin Georgiev, Khaled Zennir
Boundary Value Problems on Time Scales, Volume II
Svetlin Georgiev, Khaled Zennir
Observability and Mathematics
Fluid Mechanics, Solutions of Navier-Stokes Equations, and Modeling
Boris Khots
Handbook of Differential Equations, Fourth Edition
Daniel Zwillinger, Vladimir Dobrushkin
Experimental Statistics and Data Analysis for Mechanical and Aerospace Engineers
James Middleton
Advanced Engineering Mathematics with MATLAB•, Fifth Edition
Dean G. Duffy
Handbook of Fractional Calculus for Engineering and Science
Harendra Singh, H. M. Srivastava, Juan J. Nieto
Separation of Variables and Exact Solutions to Nonlinear PDEs
Andrei D. Polyanin, Alexei I. Zhurov
Advanced Engineering Mathematics
A Second Course with MATLAB•
Dean G. Duffy
Quantum Computation
Helmut Bez and Tony Croft
Computational Mathematics
An Introduction to Numerical Analysis and Scientific Computing with Python
Dimitrios Mitsotakis
Delay Ordinary and Partial Differential Equations
Andrei D. Polyanin, Vsevolod G. Sorkin, Alexi I. Zhurov
Clean Numerical Simulation
Shijun Liao
Multiplicative Partial Differential Equations
Svetlin Georgiev and Khaled Zennir
Engineering Statistics
A Matrix-Vector Approach with MATLAB•
Lester W. Schmerr Jr.
General Quantum Numerical Analysis
Svetlin Georgiev and Khaled Zennir
An Introduction to Partial Differential Equations with MATLAB•
Matthew P. Coleman and Vladislav Bukshtynov
Handbook of Exact Solutions to Mathematical Equations
Andrei D. Polyanin
Introducing Game Theory and its Applications, Second Edition
Elliott Mendleson and Dan Zwillinger
Modeling Operations Research and Business Analytics
William P. Fox and Robert E. Burks
https://www.routledge.com/Advances-in-Applied-Mathematics/book-series/CRCADVAPPMTH?pd=published,forthcom
ing&pg=1&pp=12&so=pub&view=list
, 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.
Solutions to Mathematical
Equations
This reference book describes the exact solutions of the following types of mathematical equa-
tions:
• Algebraic and Transcendental Equations
• Ordinary Differential Equations
• Systems of Ordinary Differential Equations
• First-Order Partial Differential Equations
• Linear Equations and Problems of Mathematical Physics
• Nonlinear Equations of Mathematical Physics
• Systems of Partial Differential Equations
• Integral Equations
• Difference and Functional Equations
• Ordinary Functional Differential Equations
• Partial Functional Differential Equations
The book pays special attention to equations found in various fields of natural and engineering
sciences (in the theory of heat and mass transfer, wave theory, hydrodynamics, gas dynamics,
combustion theory, elasticity theory, general mechanics, theoretical physics, nonlinear optics,
biology, chemical engineering sciences, ecology, and others) and equations of a reasonably gen-
eral form, which depend on free parameters or arbitrary functions.
The Handbook of Exact Solutions to Mathematical Equations is unique in world literature and
contains extensive new material. The exact solutions given in the book, being rigorous math-
ematical standards, can be used as test problems to assess the accuracy and verify the adequacy
of various numerical and approximate analytical methods for solving mathematical equations, as
well as for checking and comparing the effectiveness of exact analytical methods.
,Advances in Applied Mathematics
Series Editor: Daniel Zwillinger
Boundary Value Problems on Time Scales, Volume I
Svetlin Georgiev, Khaled Zennir
Boundary Value Problems on Time Scales, Volume II
Svetlin Georgiev, Khaled Zennir
Observability and Mathematics
Fluid Mechanics, Solutions of Navier-Stokes Equations, and Modeling
Boris Khots
Handbook of Differential Equations, Fourth Edition
Daniel Zwillinger, Vladimir Dobrushkin
Experimental Statistics and Data Analysis for Mechanical and Aerospace Engineers
James Middleton
Advanced Engineering Mathematics with MATLAB•, Fifth Edition
Dean G. Duffy
Handbook of Fractional Calculus for Engineering and Science
Harendra Singh, H. M. Srivastava, Juan J. Nieto
Separation of Variables and Exact Solutions to Nonlinear PDEs
Andrei D. Polyanin, Alexei I. Zhurov
Advanced Engineering Mathematics
A Second Course with MATLAB•
Dean G. Duffy
Quantum Computation
Helmut Bez and Tony Croft
Computational Mathematics
An Introduction to Numerical Analysis and Scientific Computing with Python
Dimitrios Mitsotakis
Delay Ordinary and Partial Differential Equations
Andrei D. Polyanin, Vsevolod G. Sorkin, Alexi I. Zhurov
Clean Numerical Simulation
Shijun Liao
Multiplicative Partial Differential Equations
Svetlin Georgiev and Khaled Zennir
Engineering Statistics
A Matrix-Vector Approach with MATLAB•
Lester W. Schmerr Jr.
General Quantum Numerical Analysis
Svetlin Georgiev and Khaled Zennir
An Introduction to Partial Differential Equations with MATLAB•
Matthew P. Coleman and Vladislav Bukshtynov
Handbook of Exact Solutions to Mathematical Equations
Andrei D. Polyanin
Introducing Game Theory and its Applications, Second Edition
Elliott Mendleson and Dan Zwillinger
Modeling Operations Research and Business Analytics
William P. Fox and Robert E. Burks
https://www.routledge.com/Advances-in-Applied-Mathematics/book-series/CRCADVAPPMTH?pd=published,forthcom
ing&pg=1&pp=12&so=pub&view=list
, 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.
