# Differential and Difference Equations: A Comparison of Methods of Solution

Springer, Apr 18, 2016 - Science - 162 pages

This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, reduction of order, Laplace transforms and generating functions - bringing out the similarities as well as the significant differences in the respective analyses. Equations of arbitrary order are studied, followed by a detailed analysis for equations of first and second order. Equations with polynomial coefficients are considered and explicit solutions for equations with linear coefficients are given, showing significant differences in the functional form of solutions of differential equations from those of difference equations. An alternative method of solution involving transformation of both the dependent and independent variables is given for both differential and difference equations. A comprehensive, detailed treatment of Green’s functions and the associated initial and boundary conditions is presented for differential and difference equations of both arbitrary and second order. A dictionary of difference equations with polynomial coefficients provides a unique compilation of second order difference equations obeyed by the special functions of mathematical physics. Appendices augmenting the text include, in particular, a proof of Cramer’s rule, a detailed consideration of the role of the superposition principal in the Green’s function, and a derivation of the inverse of Laplace transforms and generating functions of particular use in the solution of second order linear differential and difference equations with linear coefficients.

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### Contents

 1 Operators 2 2 Solution of Homogeneous and Inhomogeneous Linear Equations 5 3 First Order Homogeneous and Inhomogeneous Linear Equations 23 4 Second Order Homogeneous and Inhomogeneous Equations 27 5 Selfadjoint Linear Equations 35 6 Greens Function 39 7 Generating Functions ZTransforms Laplace Transforms and the Solution of Linear Differential and Difference Equations 63 8 Dictionary of Difference Equations with Polynomial Coefficients 112
 Appendix D Casoratian Determinant 137 Appendix E Cramers Rule 139 Appendix F Greens Function and the SuperpositionPrinciple 141 Appendix G Inverse Laplace Transforms and InverseGenerating Functions 144 Appendix H Hypergeometric Function 149 Appendix I Confluent Hypergeometric Functions 151 Appendix J Solutions of the Second Kind 155 Bibliography 159

 Appendix A Difference Operator 125 Appendix B Notation 130 Appendix C Wronskian Determinant 133
 Index 161 Copyright

### About the author (2016)

Leonard Maximon is Research Professor of Physics in the Department of Physics at The George Washington University and Adjunct Professor in the Department of Physics at Arizona State University. He has been an Assistant Professor in the Graduate Division of Applied Mathematics at Brown University, a Visiting Professor at the Norwegian Technical University in Trondheim, Norway, and a Physicist at the Center for Radiation Research at the National Bureau of Standards. He is also an Associate Editor for Physics for the DLMF project and a Fellow of the American Physical Society.

Maximon has published numerous papers on the fundamental processes of quantum electrodynamics and on the special functions of mathematical physics.