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1. Introduction to Differential Equations
1.1 Definitions and Terminology
1.2 Initial-Value Problems
1.3 Differential Equations as Mathematical Models
2. First-Order Differential Equations
2.1 Solution Curves Without a Solution
- 2.1.1 Direction Fields
- 2.1.2 Autonomous First-Order DEs
2.2 Separable Variables
2.3 Linear Equations
2.4 Exact Equations
2.5 Solutions by Substitutions
2.6 A Numerical Method
3. Modeling with First-Order Differential Equations
3.1 Linear Models
3.2 Nonlinear Models
3.3 Modeling with Systems of First-Order DEs
4. Higher-Order Differential Equations
4.1 Preliminary Theory—Linear Equations
- 4.1.1 Initial-Value and Boundary-Value Problems
- 4.1.2 Homogeneous Equations
- 4.1.3 Nonhomogeneous Equations
4.2 Reduction of Order
4.3 Homogeneous Linear Equations with Constant Coefficients
4.4 Undetermined Coefficients—Superposition Approach
4.5 Undetermined Coefficients—Annihilator Approach
4.6 Variation of Parameters
4.7 Cauchy-Euler Equation
4.8 Solving Systems of Linear DEs by Elimination
4.9 Nonlinear Differential Equations
5. Modeling with Higher-Order Differential Equations
5.1 Linear Models: Initial-Value Problems
- 5.1.1 Spring/Mass Systems: Free Undamped Motion
- 5.1.2 Spring/Mass Systems: Free Damped Motion
- 5.1.3 Spring/Mass Systems: Driven Motion
- 5.1.4 Series Circuit Analogue
5.2 Linear Models: Boundary-Value Problems
5.3 Nonlinear Models
6. Series Solutions of Linear Equations
6.1 Solutions About Ordinary Points
- 6.1.1 Review of Power Series
- 6.1.2 Power Series Solutions
6.2 Solutions About Singular Points
6.3 Special Functions
- 6.3.1 Bessel’s Equation
- 6.3.2 Legendre’s Equation
7. The Laplace Transform
7.1 Definition of the Laplace Transform
7.2 Inverse Transforms and Transforms of Derivatives
- 7.2.1 Inverse Transforms
- 7.2.2 Transforms of Derivatives
7.3 Operational Properties I
- 7.3.1 Translation on the s-Axis
- 7.3.2 Translation on the t-Axis
7.4 Operational Properties II
- 7.4.1 Derivatives of a Transform
- 7.4.2 Transforms of Integrals
- 7.4.3 Transform of a Periodic Function
7.5 The Dirac Delta Function
7.6 Systems of Linear Differential Equations
8. Systems of Linear First-Order Differential Equations
8.1 Preliminary Theory—Linear Systems
8.2 Homogeneous Linear Systems
- 8.2.1 Distinct Real Eigenvalues
- 8.2.2 Repeated Eigenvalues
- 8.2.3 Complex Eigenvalues
8.3 Nonhomogeneous Linear Systems
- 8.3.1 Undetermined Coefficients
- 8.3.2 Variation of Parameters
8.4 Matrix Exponential
9. Numerical Solutions of Ordinary Differential Equations
9.1 Euler Methods and Error Analysis
9.2 Runge-Kutta Methods
9.3 Multistep Methods
9.4 Higher-Order Equations and Systems
9.5 Second-Order Boundary-Value Problems
10. Plane Autonomous Systems
10.1 Autonomous Systems
10.2 Stability of Linear Systems
10.3 Linearization and Local Stability
10.4 Autonomous Systems as Mathematical Models
11. Orthogonal Functions and Fourier Series
11.1 Orthogonal Functions
11.2 Fourier Series
11.3 Fourier Cosine and Sine Series
11.4 Sturm-Liouville Problem
11.5 Bessel and Legendre Series
- 11.5.1 Fourier-Bessel Series
- 11.5.2 Fourier-Legendre Series
12. Boundary-Value Problems in Rectangular Coordinates
12.1 Separable Partial Differential Equations
12.2 Classical PDEs and Boundary-Value Problems
12.3 Heat Equation
12.4 Wave Equation
12.5 Laplace’s Equation
12.6 Nonhomogeneous Boundary-Value Problems
12.7 Orthogonal Series Expansions
12.8 Higher-Dimensional Problems
