Differential Equations: Modeling with MATLAB, 1/e
Published April, 1999 by Prentice Hall Engineering/Science/Mathematics
Copyright 1999, 641 pp.
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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 instructorfor
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
Most ideas are introduced in a spiral to build intuition
or to supply motivation before they are formally introduced.
- To encourage self-study, each exercise set includes a
guide to steer students toward areas in which they need practice.
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
5. Two-Dimensional Models: Oscillating Systems.
Overviewpopulations, 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
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.
Solutions to Selected Exercises
MATLAB tutorial. Calculus review.