Differential Equations: Theory and Applications

425 Pages·2018·2.347 MB·English

by  Maria Catherine Borres

DIFFERENTIAL EQUATIONS:
THEORY AND APPLICATIONS

DIFFERENTIAL EQUATIONS: THEORY
AND APPLICATIONS
Maria Catherine Borres
ARCLER
P r e ss
www.arclerpress.com
Differential Equations: Theory and Applications
Maria Catherine Borres
Arcler Press
2010 Winston Park Drive,
2nd Floor
Oakville, ON L6H 5R7
Canada
www.arclerpress.com
Tel: [phone]
001-905-616-2116
Fax: [phone]
Email: [email protected]
e-book Edition 2019
ISBN: 978-1-77361-596-7 (e-book)
This book contains information obtained from highly regarded resources. Reprinted material
sources are indicated and copyright remains with the original owners. Copyright for images and
other graphics remains with the original owners as indicated. A Wide variety of references are
listed. Reasonable efforts have been made to publish reliable data. Authors or Editors or Publish-
ers are not responsible for the accuracy of the information in the published chapters or conse-
quences of their use. The publisher assumes no responsibility for any damage or grievance to the
persons or property arising out of the use of any materials, instructions, methods or thoughts in
the book. The authors or editors and the publisher have attempted to trace the copyright holders
of all material reproduced in this publication and apologize to copyright holders if permission has
not been obtained. If any copyright holder has not been acknowledged, please write to us so we
may rectify.
Notice: Registered trademark of products or corporate names are used only for explanation and
identification without intent of infringement.
© 2019 Arcler Press
ISBN: 978-1-77361-403-8 (Hardcover)
Arcler Press publishes wide variety of books and eBooks. For more information about
Arcler Press and its products, visit our website at www.arclerpress.com
ABOUT THE AUTHOR
Catherine is currently taking up Master of Arts in Education Major in
Mathematics in Philippine Normal University – Manila. She is currently
working as a Content Developer for Mathematics at the Affordable Private
Education Center (APEC Schools).

TABLE OF CONTENTS
Preface........................................................................ ..................................xi
Chapter 1 Basic Concepts of Differential Equations ...................................................1
1.1. Introduction ........................................................................................2
1.2. The Bernoulli Equation ......................................................................31
1.3. Differential Equations of Higher Order ..............................................50
1.4. The Wronskian ..................................................................................85
Chapter 2 Fundamental Concepts of Partial Differential Equations .......................111
2.1. Introduction ....................................................................................112
2.2. Classification of Second Order PDE ...............................................112
2.3. Summary and Discussion ................................................................141
2.4. Classification of Second Order PDE ...............................................141
Chapter 3 Application of Differential Equations In Mechanics ..............................143
3.1. Introduction ....................................................................................144
3.2. Projectile Motion ...........................................................................161
3.3. Summary and Discussion ................................................................186
Chapter 4 Elliptic Differential Equation .................................................................191
4.1. Introduction ....................................................................................192
4.2. Boundary Value Problem (BVPs) .....................................................195
4.3. Some Important Mathematical Tools ...............................................197
4.4. Properties Of Harmonic Functions ..................................................199
4.5. Separation Of Variables ..................................................................210
4.6. Dirichlet Problem For A Rectangle ..................................................212
4.7. The Neumann Problem For A Rectangle .........................................215
4.8. Interior Dirichlet Problem For A Circle ...........................................217
4.9. Exterior Dirichlet Problem For A Circle ..........................................222
4.10. Interior Neumann Problem For A Circle ........................................227
4.11. Solution Of Laplace Equation In Cylindrical Coordinates .............229
4.12. Solution Of Laplace Equation In Spherical Coordinates ................238
4.13. Miscellaneous Example ................................................................247
4.14. Summary And Discussions ............................................................276
Chapter 5 Hyperbolic Differential Equation ..........................................................279
5.1 Introduction .....................................................................................280
5.2. Solution Of One-Dimensional Wave Equation by
Canonical Reduction ...................................................................284
5.3. The Initial Value Problem; D’alembert’s Solution .............................288
5.4. Summary And Discussion ...............................................................297
Chapter 6 Parabolic Differential Equations ............................................................301
6.1. Introduction ....................................................................................302
6.2. Boundary Conditions ......................................................................304
6.3. Elementary Solutions Of The Diffusion Equation ............................305
6.4. Dirac Delta Function ......................................................................310
6.5. Separation Of Variables Method ......................................................316
6.6. Maximum-Minimum Principle and Consequences ..........................340
6.7. Miscellaneous Example. ................................................................343
6.8. Boundary Conditions ......................................................................352
Chapter 7 Laplace Transform Methods ..................................................................357
7.1. Introduction ....................................................................................358
7.2. Transform Of Some Elementary Functions .......................................362
7.3. Properties Of Laplace Transform .....................................................364
7.4. Transform Of A Periodic Function ...................................................370
7.5. Transform Of Error Function ............................................................372
7.6. Transform Of Bessel’s Function .......................................................374
7.7. Transform Of Dirac Delta Function .................................................376
7.8. Convolution Theorem (Faltung Theorem) .........................................382
Chapter 8 Green’s Function ...................................................................................389
8.1. Introduction ....................................................................................390
8.2. The Eigenfunction Method ..............................................................403
viii