Differential Equations and Linear Algebra – Edwards & Penney – 3rd Edition

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 Second-Order Systems and Mechanical Applications

7.5 Multiple Eigenvalue Solutions

7.6 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

References for Further Study

Appendix A: Existence and Uniqueness of Solutions

Appendix B: Theory of Determinants

Answers to Selected Problems

Index