Skip to main content Skip to main navigation


For courses in Differential Equations and Linear Algebra.


The right balance between concepts, visualization, applications, and skills 

Differential Equations and Linear Algebra provides the conceptual development and geometric visualization of a modern differential equations and linear algebra course that is essential to science and engineering students. It balances traditional manual methods with the new, computer-based methods that illuminate qualitative phenomena — a comprehensive approach that makes accessible a wider range of more realistic applications.

The book combines core topics in elementary differential equations with concepts and methods of elementary linear algebra. It starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout. 

For the first time, MyLab™ Math is available for this text, providing online homework with immediate feedback, the complete eText, and more. Additionally, new presentation slides created by author David Calvis are available in Beamer (LaTeX) and PDF formats. The slides are ideal for classroom lectures and student review, and combined with Calvis’ superlative instructional videos offer a level of support not found in any other Differential Equations course.

Also available with MyLab Math

MyLab Math is the teaching and learning platform that empowers you to reach every student. By combining trusted author content with digital tools and a flexible platform, MyLab Math personalizes the learning experience and improves results for each student.



Table of contents

1. First-Order Differential Equations

1.1 Differential Equations and Mathematical Models

1.2 Integrals as General and Particular Solutions

1.3 Slope Fields and Solution Curves

1.4 Separable Equations and Applications

1.5 Linear First-Order Equations

1.6 Substitution Methods and Exact Equations


2. Mathematical Models and Numerical Methods

2.1 Population Models

2.2 Equilibrium Solutions and Stability

2.3 Acceleration–Velocity Models

2.4 Numerical Approximation: Euler's Method

2.5 A Closer Look at the Euler Method

2.6 The Runge–Kutta Method


3. Linear Systems and Matrices

3.1 Introduction to Linear Systems

3.2 Matrices and Gaussian Elimination

3.3 Reduced Row-Echelon Matrices

3.4 Matrix Operations

3.5 Inverses of Matrices

3.6 Determinants

3.7 Linear Equations and Curve Fitting


4. Vector Spaces

4.1 The Vector Space R3

4.2 The Vector Space Rn and Subspaces

4.3 Linear Combinations and Independence of Vectors

4.4 Bases and Dimension for Vector Spaces

4.5 Row and Column Spaces

4.6 Orthogonal Vectors in Rn

4.7 General Vector Spaces


5. Higher-Order Linear Differential Equations

5.1 Introduction: Second-Order Linear Equations

5.2 General Solutions of Linear Equations

5.3 Homogeneous Equations with Constant Coefficients

5.4 Mechanical Vibrations

5.5 Nonhomogeneous Equations and Undetermined Coefficients

5.6 Forced Oscillations and Resonance


6. Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues

6.2 Diagonalization of Matrices

6.3 Applications Involving Powers of Matrices


7. Linear Systems of Differential Equations

7.1 First-Order Systems and Applications

7.2 Matrices and Linear Systems

7.3 The Eigenvalue Method for Linear Systems

7.4 A Gallery of Solution Curves of Linear Systems

7.5 Second-Order Systems and Mechanical Applications

7.6 Multiple Eigenvalue Solutions

7.7 Numerical Methods for Systems


8. Matrix Exponential Methods

8.1 Matrix Exponentials and Linear Systems

8.2 Nonhomogeneous Linear Systems

8.3 Spectral Decomposition Methods


9. Nonlinear Systems and Phenomena

9.1 Stability and the Phase Plane

9.2 Linear and Almost Linear Systems

9.3 Ecological Models: Predators and Competitors

9.4 Nonlinear Mechanical Systems


10. Laplace Transform Methods

10.1 Laplace Transforms and Inverse Transforms

10.2 Transformation of Initial Value Problems

10.3 Translation and Partial Fractions

10.4 Derivatives, Integrals, and Products of Transforms

10.5 Periodic and Piecewise Continuous Input Functions


11. Power Series Methods

11.1 Introduction and Review of Power Series

11.2 Power Series Solutions

11.3 Frobenius Series Solutions

11.4 Bessel Functions


Appendix A: Existence and Uniqueness of Solutions

Appendix B: Theory of Determinants




The modules listed below follow the indicated sections in the text. Most provide computing projects that illustrate the corresponding text sections. Many of these modules are enhanced by the supplementary material found at the new Expanded Applications website.


1.3 Computer-Generated Slope Fields and Solution Curves

1.4 The Logistic Equation

1.5 Indoor Temperature Oscillations

1.6 Computer Algebra Solutions

2.1 Logistic Modeling of Population Data

2.3 Rocket Propulsion

2.4 Implementing Euler's Method

2.5 Improved Euler Implementation

2.6 Runge-Kutta Implementation

3.2 Automated Row Operations

3.3 Automated Row Reduction

3.5 Automated Solution of Linear Systems

5.1 Plotting Second-Order Solution Families

5.2 Plotting Third-Order Solution Families

5.3 Approximate Solutions of Linear Equations

5.5 Automated Variation of Parameters

5.6 Forced Vibrations and Resonance

7.1 Gravitation and Kepler's Laws of Planetary Motion

7.3 Automatic Calculation of Eigenvalues and Eigenvectors

7.4 Dynamic Phase Plane Graphics

7.5 Earthquake-Induced Vibrations of Multistory Buildings

7.6 Defective Eigenvalues and Generalized Eigenvectors

7.7 Comets and Spacecraft

8.1 Automated Matrix Exponential Solutions

8.2 Automated Variation of Parameters

9.1 Phase Portraits and First-Order Equations

9.2 Phase Portraits of Almost Linear Systems

9.3 Your Own Wildlife Conservation Preserve

9.4 The Rayleigh and van der Pol Equations

For teachers

All the material you need to teach your courses.

Discover teaching material