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Differential Equations with Linear Algebra

M. R. Boelkins, M. C. Potter, and J. L. Goldberg
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Tom Schulte
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This text starts from rate problems at the level of a college freshman or advanced high school student, such as solutions mixing between connected tanks. This sets up the preliminaries for systems of linear equations. The content moves from there to the basics of linear algebra to include matrices, eigenvalues, and vector spaces. From a similar starting level, Chapter Two introduces first-order ODEs, including both linear and non-linear equations, separable and exact equations, and applications from mixing problems, growth, cooling, logistics, and Torricelli’s Law. With these chapters, Differential Equations with Linear Algebra works as an efficiently self-contained and effectively presented introduction of linear systems of DEs that also takes the reader to higher order and nonlinear systems of DEs, Laplace Transforms, numerical methods, and series solutions for DEs.

The book engages the reader and supports comprehension through a consistently applied chapter framework. Each chapter begins with motivating topics that are then answered with introduced theorems and techniques. Nonlinear equations naturally motivate for numerical methods while the versatile conjoined tanks provide motivation for first-order ODEs and linear systems of DEs. Sections include exercises with odd-numbered solutions. Applications from computer graphics to damped motion bring the theory into the real world and implementation is suggested in many Maple examples, and a few for Excel. Maple examples include direction fields for nonlinear systems, Laplace transforms, and more. Excel examples include Euler’s method.

Linearity is a sturdy thread that unites differential equations with linear algebra here. This book does the merging of topics very well and requires only a first semester in multivariable calculus as an entry point. The application- and implementation-rich approach will work well for future applied mathematics, scientists, and engineers.

Tom Schulte teaches at Michigan’s Oakland Community College and graduated from Oakland University. He is still heard at OU hosting a nonlinear program on the campus station, WXOU.

1. Essentials of Linear Algebra
2. First-Order Differential Equations
3. Linear Systems of Differential Equations
4. Higher-Order Differential Equations
5. Laplace Tranforms
6. Nonlinear Systems of Differential Equations
7. Numerical Methods for Differential Equations
8. Series Solutions for Differential Equations
A. Review of Integration Techniques
B. Complex Numbers
C. Roots of Polynomials
D. Linear Transformations
E. Solutions to Selected Exercises