A.I

A.II

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APPENDIX I

GAMMA FUNCTION

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APPENDIX II

MATRICES

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1

Introduction to Differential Equations

1.R

1.1

Definitions and Terminology

1.1

Definitions and Terminology

1.2

Initial-Value Problems

1.2

Initial-Value Problems

1.3

Differential Equations as Mathematical Models

1.3

Differential Equations as Mathematical Models

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2

First-order Differential Equations

2.R

2.1

Solution Curves Without a Solution

2.1

Solution Curves Without a Solution

2.2

Separable Equations

2.2

Separable Equations

2.3

Linear Equations

2.3

Linear Equations

2.4

Exact Equations

2.4

Exact Equations

2.5

Solutions by Substitutions

2.5

Solutions by Substitutions

2.6

A Numerical Method

2.6

A Numerical Method

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3

Modeling with First-Order Differential Equations

3.R

3.1

Linear Models

3.1

Linear Models

3.2

Nonlinear Models

3.2

Nonlinear Models

3.3

Modeling with Systems of First-Order DEs

3.3

Modeling with Systems of First-Order DEs

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4

Higher-Order Differential Equations

4.R

4.1

Preliminary Theory—Linear Equations

4.1

Preliminary Theory—Linear Equations

4.10

Nonlinear Differential Equations

4.2

Reduction of Order

4.2

Reduction of Order

4.3

Homogeneous Linear Equations with Constant Coefficient

4.3

Homogeneous Linear Equations with Constant Coefficient

4.4

Undetermined Coefficients—Superposition Approach

4.4

Undetermined Coefficients—Superposition Approach

4.5

Undetermined Coefficients—Annihilator Approach

4.5

Undetermined Coefficients—Annihilator Approach

4.6

Variation of Parameters

4.6

Variation of Parameters

4.7

Cauchy-Euler Equation

4.7

Cauchy-Euler Equation

4.8

Green’s Functions

4.8

Green’s Functions

4.9

Solving Systems of Linear DEs by Elimination

4.9

Solving Systems of Linear DEs by Elimination

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5

Modeling with Higher-Order Differential Equations

5.R

5.1

Linear Models: Initial-Value Problems

5.1

Linear Models: Initial-Value Problems

5.2

Linear Models: Boundary-Value Problems

5.2

Linear Models: Boundary-Value Problems

5.3

Nonlinear Models

5.3

Nonlinear Models

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6

Series Solutions of Linear Equations

6.R

6.1

Review of Power Series

6.1

Review of Power Series

6.2

Solutions About Ordinary Points

6.2

Solutions About Ordinary Points

6.3

Solutions About Singular Points

6.3

Solutions About Singular Points

6.4

Special Functions

6.4

Special Functions

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7

The Laplace Transform

7.R

7.1

Definition of the Laplace Transform

7.1

Definition of the Laplace Transform

7.2

Inverse Transforms and Transforms of Derivatives

7.2

Inverse Transforms and Transforms of Derivatives

7.3

Operational Properties I

7.3

Operational Properties I

7.4

Operational Properties II

7.4

Operational Properties II

7.5

The Dirac Delta Function

7.5

The Dirac Delta Function

7.6

Systems of Linear Differential Equations

7.6

Systems of Linear Differential Equations

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8

Systems of Linear First-Order Differential Equations

8.R

8.1

Preliminary Theory—Linear Systems

8.1

Preliminary Theory—Linear Systems

8.2

Homogeneous Linear Systems

8.2

Homogeneous Linear Systems

8.3

Nonhomogeneous Linear Systems

8.3

Nonhomogeneous Linear Systems

8.4

Matrix Exponential

8.4

Matrix Exponential

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9

Numerical Solutions of Ordinary Differential Equations

9.R

9.1

Euler Methods and Error Analysis

9.1

Euler Methods and Error Analysis

9.2

Runge-Kutta Methods

9.2

Runge-Kutta Methods

9.3

Multistep Methods

9.3

Multistep Methods

9.4

Higher-Order Equations and Systems

9.4

Higher-Order Equations and Systems

9.5

Second-Order Boundary-Value Problems

9.5

Second-Order Boundary-Value Problems