## Differential Equations: Modeling with MATLAB, 1/e

Paul Davis

Published April, 1999 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-736539-X

mailings
on this subject.

Differential Equations-Mathematics

For undergraduate engineering and science courses in Differential Equations. This progressive text on differential equations utilizes MATLAB's state-of-the-art computational and graphical tools right from the start to help students probe a variety of mathematical models. Ideas are examined from four perspectives: geometric, analytic, numeric, and physical. Students are encouraged to develop problem-solving skills and independent judgment as they derive models, select approaches to their analysis, and find answers to the original, physical questions. Both qualitative and algebraic tools are stressed.

Balancing the qualitative with the algebraic, the text exposes students in the first two chapters to fundamental qualitative ideas such as direction fields, steady states, stability, etc. Then graphical interpretation, analytic solutions, and numerical tools are developed to allow students to examine nonlinear problems and systems of equations. This is done in conjunction with covering the most important traditional, analytic methods.
Supports student reading with numerous examples, MATLAB exercises, and thought questions woven seamlessly into the text. Thought questions are noted as such for easy reference by the instructor—for use in class discussion or as a homework assignment.
Students may purchase the student edition of MATLAB at a specially discounted price packaged with the text.
Many exercises are posed from the physical perspective of the models under study in order to nurture students' ability to easily shift between theoretical/mathematical and real/physical settings.
Provides students with more than 1,400 exercises covering a carefully graded spectrum from routine, confidence-building questions to more open-ended challenges that invite interpretation and deeper analysis.

• To encourage self-study, each exercise set includes a guide to steer students toward areas in which they need practice.
Most ideas are introduced in a spiral to build intuition or to supply motivation before they are formally introduced.

1. Prologue.

Goals. A modeling example. Differential equations and solutions.

2. Models from Conservation Laws.

Simple population growth. Emigration and competition. Heat flow. Multiple species.

3. Numerical and Graphical Tools.

Numerical methods. Graphs, direction fields, and phase lines. Steady states, stability, and local linearization.

4. Analytic Tools for One Dimension.

Basic definitions. Separation of variables. Characteristic equations. Undetermined coefficients. Variation of parameters. Existence and uniqueness.

5. Two-Dimensional Models: Oscillating Systems.

Overview—populations, position, and velocity. Spring-mass systems. Pendulum. RLC circuits.

6. Analytic Tools for Two Dimensions:

Basic definitions. The Wronskian and linear independence. Characteristic equations: real roots. Characteristic equations: complex roots. Unforced spring-mass systems. Undetermined coefficients. Forced spring-mass systems. Linear vs. nonlinear.

7. Graphical Tools for Two Dimensions.

Back to the phase plane. More phase plane: nullclines, steady states, stability. Limit cycles.

8. Analytic Tools for Higher Dimensions: Systems.

Motivation and review. Basic definitions. Homogeneous systems. Connections with the phase plane. Nonhomogeneous systems: undetermined coefficients.

9. Diffusion Models and Boundary-Value Problems.

Diffusion models. Boundary-value problems. Buckling. Time-dependent diffusion. Fourier methods. Numerical tools: time-dependent diffusion. Finite difference approximations to steady states.

10. Laplace Transform.

The transform idea: jumps and filters. Inverse transforms. Other properties of Laplace transforms. Ramps and jumps. The unit impulse function. Control applications.

11. More Analytic Tools for Two Dimensions..

Variation of parameters for systems. Variation of parameters for second-order equations. Reduction of order. Cauchy-Euler equations. Power series methods. Regular singular points. Solution method summary.

Appendicies.
Bibliography
Solutions to Selected Exercises
Index

MATLAB tutorial. Calculus review.