# Differential and Difference Equations

## A Comparison of Methods of Solution

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. Les mer
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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.

Preface.

Introduction.

1 Operators.

2 Solution of homogeneous and inhomogeneous linear
equations.

2.1 Variation of constants. 2.2 Reduction of order when one solution to the
homogeneous equation is known.

3 First order homogeneous and inhomogeneous linear
equations.

4 Second-order homogeneous and inhomogeneous equations.

5 Self-adjoint linear equations.

6 Green's function.

6.1 Differential equations. 6.2 Difference equations.

7 Generating function, z-transforms, Laplace transforms and
the solution of linear differential and difference equations.

7.1 Laplace transforms and the solution of linear differential equations with constant coefficients. 7.2 Generating
functions and the solution of linear difference equations with constant
coefficient. 7.3 Laplace transforms and the solution of linear differential
equations with polynomial coefficients. 7.4 Alternative method for the solution
of homogeneous linear differential equations with linear coefficients. 7.5 Generating functions and the solution of
linear difference equations with polynomial coefficients. 7.6 Solution of
homogeneous linear difference equations with linear coefficients.

8 Dictionary of difference equations with polynomial
coefficients.

Appendix A: Difference operator.

Appendix B: Notation.

Appendix C: Wronskian Determinant.

Appendix D: Casoratian Determinant.

Appendix E: Cramer's Rule.

Appendix F: Green's function and the Superposition
principle.

Appendix G: Inverse Laplace transforms and Inverse
Generating functions.

Appendix H: Hypergeometric function.

Appendix I: Confluent
Hypergeometric function.

Appendix J. Solutions of the second kind.

Bibliography.

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 