Differential Equations: An Introduction to Modern Methods and Applications – William E. Boyce – 2nd Edition

1 Introduction.

1.1 Mathematical Models, Solutions, and Direction Fields.
1.2 Linear Equations: Method of Integrating Factors.
1.3 Numerical Approximations: Euler’s Method.
1.4 Classification of Differential Equations.

2 First Order Differential Equations.
2.1 Separable Equations.
2.2 Modeling with First Order Equations.
2.3 Differences Between Linear and Nonlinear Equations.
2.4 Autonomous Equations and Population Dynamics.
2.5 Exact Equations and Integrating Factors.
2.6 Accuracy of Numerical Methods.
2.7 Improved Euler and Runge–Kutta Methods.

3 Systems of Two First Order Equations.
3.1 Systems of Two Linear Algebraic Equations.
3.2 Systems of Two First Order Linear Differential Equations.
3.3 Homogeneous Linear Systems with Constant Coefficients.
3.4 Complex Eigenvalues.
3.5 Repeated Eigenvalues.
3.6 A Brief Introduction to Nonlinear Systems.
3.7 Numerical Methods for Systems of First Order Equations.

4 Second Order Linear Equations.
4.1 Definitions and Examples.
4.2 Theory of Second Order Linear Homogeneous Equations.
4.3 Linear Homogeneous Equations with Constant Coefficients.
4.4 Mechanical and Electrical Vibrations.
4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients.
4.6 Forced Vibrations, Frequency Response, and Resonance.
4.7 Variation of Parameters.

5 The Laplace Transform.
5.1 Definition of the Laplace Transform.
5.2 Properties of the Laplace Transform.
5.3 The Inverse Laplace Transform.
5.4 Solving Differential Equations with Laplace Transforms.
5.5 Discontinuous Functions and Periodic Functions.
5.6 Differential Equations with Discontinuous Forcing Functions.
5.7 Impulse Functions.
5.8 Convolution Integrals and Their Applications.
5.9 Linear Systems and Feedback Control.

6 Systems of First Order Linear Equations.
6.1 Definitions and Examples.
6.2 Basic Theory of First Order Linear Systems.
6.3 Homogeneous Linear Systems with Constant Coefficients.
6.4 Nondefective Matrices with Complex Eigenvalues.
6.5 Fundamental Matrices and the Exponential of a Matrix.
6.6 Nonhomogeneous Linear Systems.
6.7 Defective Matrices.

7 Nonlinear Differential Equations and Stability.
7.1 Autonomous Systems and Stability.
7.2 Almost Linear Systems.
7.3 Competing Species.
7.4 Predator–Prey Equations.
7.5 Periodic Solutions and Limit Cycles.
7.6 Chaos and Strange Attractors: The Lorenz Equations.

Apendix
A Matrices and Linear Algebra.
A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank.
A.3 Determinants and Inverses.
A.4 The Eigenvalue Problem.