For courses in Differential Equations and Linear Algebra.

Known for its real-world applications and its blend of algebraic and geometric approaches, this text offers a full treatment of differential equations together with the linear algebra topics that students need. It emphasizes the conceptual development and geometric visualization of reform courses, but retains the solid foundation of symbolic techniques that remain important to science and engineering students. The authors frame the text by discussing mathematical modeling of real-world phenomena, with a fresh new computational and qualitative flavor evident throughout in figures, examples, problems, and applications. Scientific computing environments like Maple, Mathematica, and MATLAB are available in a separate manual (FREE if shrink-wrapped with text).

Chapter 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.

Chapter 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

Chapter 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.

Chapter 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.

Chapter 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: Undetermined Coefficients and Variation of Parameters. 5.6: Forced Oscillations and Resonance.

Chapter 6: Eigenvalues and Eigenvectors

6.1: Introduction to Eigenvalues. 6.2: Diagonalization of Matrices. 6.3: Applications Involving Powers of Matrices

Chapter 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: Second-Order Systems and Mechanical Applications. 7.5: Multiple Eigenvalue Solutions. 7.6: Numerical Methods for Systems.

Chapter 8: Matrix Exponential Methods

8.1: Matrix Exponentials and Linear Systems. 8.2: Nonhomogeneous Linear Systems. 8.3: Spectral Decomposition Methods.

Chapter 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.

Chapter 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.

Chapter 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.
References for Further Study.
Appendix A: Existence and Uniqueness of Solutions.
Appendix B:Theory of Determinants.
Answers to Selected Problems.