(NOTE: Each chapter begins with an Introduction and concludes with a Review.)
Preface.
1. Introduction to Differential Equations.
Definitions and Terminology.
2. Linear First-Order Equations.
Homogeneous Case; Solution by Separation of Variables. Nonhomogeneous Case. Applications of Linear First-Order Equations.
3. General First-Order Equations.
Separable Equations. Existence and Uniqueness. Exact Equations and Ones That Can Be Made Exact. Additional Applications. Linear and Nonlinear Equations Contrasted.
4. Vectors and n -Space.
Geometrical Representation of "Arrow" Vectors. n-Space. Dot Product, Norm, and Angle for n-Space. Gauss Elimination. Span. Linear Dependence and Independence. Vector Space. Bases and Expansions.
5. Matrices and Linear Algebraic Equations.
Matrices and Matrix Algebra. The Transpose Matrix. Determinants. The Rank of a Matrix. Inverse Matrix, Cramer's Rule, and Factorization. Existence and Uniqueness for the System Ax = c. Vector Transformation (Optional).
6. Linear Differential Equations of Second Order and Higher.
The Complex Plane and the Exponential, Trigonometric, and Hyperbolic Functions. Linear Dependence and Linear Independence of Functions. Homogeneous Equation; General Solution. Homogeneous Equations with Constant Coefficients. Homogeneous Equations with Nonconstant Coefficients; Cauchy-Euler Equation. Solution of Nonhomogeneous Equation.
7. Applications of Linear Constant-Coefficient Equations.
Linear Harmonic Oscillator; Free Oscillation. Linear Harmonic Oscillator; Phase Plane. Linear Harmonic Oscillator; Forced Oscillation. Additional Applications.
8. Power Series Solution.
Power Series. Power Series Solutions. Introduction to the Singular Case.
9. The Eigenvalue Problem.
Solution of the Eigenvalue Problem. The Special Case of Symmetric Matrices. Differential Equation Boundary Value Problems as Eigenvalue Problems.
10. Systems of Linear Differential Equations.
Existence, Uniqueness, and General Solution. Solution by Elimination; Constant-Coefficient Equations. Solution of Homogeneous Systems as Eigenvalue Problems. Diagonalization (Optional). Triangularization (Optional). Explicit Solution of x' = Ax and the Matrix Exponential Function (Optional).
11. Quantitative Methods: The Phase Plane.
The Phase Plane. Singular Point Analysis. Additional Applications.
12. Qualitative Methods: Numerical Solution.
Euler's Method. Improvements: Runge-Kutta Methods. Application to Systems and Boundary Value Problems.
13. Laplace Transform.
Definitions and Calculation of the Transform. Properties of the Transform. Application to the Solution of Differential Equations. Discontinuous Forcing Functions; Heaviside Step Function. Additional Properties.
Appendix.
Answers to Selected Exercises.
Index.
