A Workbook for Differential Equations

1 Modeling with Differential Equations.

1.1 Terminology.

1.2 Differential Equations Describing Populations.

1.3 Remarks on Modeling with Differential Equations.

1.4 Newton's Law of Cooling.

1.5 Loaded Horizontal Beams.

2 Some Special First Order Ordinary Differential Equations.

2.1 Separable Differential Equations.

2.2 Linear First Order Differential Equations.

2.3 Bernoulli Equations.

2.4 Homogeneous Equations.

2.5 Exact Differential Equations.

2.6 Projects.

2.7 How to Review and Remember.

2.8 Review of First Order Differential Equations.

Before Module 3 Oscillating Systems and Hanging Cables.

B3.1 Spring-Mass-Systems.

B3.2 LRC Circuits.

B3.3 The Simple Pendulum.

B3.4 Suspended Cables.

B3.5 Projects.

3 Linear Differential Equations with Constant Coefficients.

3.1 Homogeneous Linear Differential Equations with Constant Coefficients.

3.2 Solving Initial and Boundary Value Problems.

3.3 Designing Oscillating Systems.

3.4 The Method of Undetermined Coefficients.

3.5 Variation of Parameters.

3.6 Cauchy-Euler Equations.

3.7 Some Results on Boundary Value Problems.

3.8 Projects.

4 Qualitative and Numerical Analysis of Differential Equations.

4.1 Direction Fields and Autonomous Equations.

4.2 From Visualization to Algorithm: Euler's Method.

4.3 Runge-Kutta Methods.

4.4 Finite Difference Methods for Second Order Boundary Value Problems.

5 Linear Differential Equations-Theory.

5.1 Existence and Uniqueness of Solutions.

5.2 Linear Independence for Vectors.

5.3 Matrices and Determinants.

5.4 Linear Independence for Functions.

5.5 The General Solution of Homogeneous Equations.

Before Module 6 Coupled Electrical and Mechanical Systems.

B6.1 Multi-Loop Circuits and Kirchhoff's Laws.

B6.2 Coupled Spring-Mass-Systems.

6 Laplace Transforms.

6.1 Introducing the Laplace Transform.

6.2 Solving Differential Equations with Laplace Transforms.

6.3 Systems of Linear Differential Equations.

6.4 Expanding the Transform Table.

6.5 Discontinuous Forcing Terms.

6.6 Complicated Forcing Functions and Convolutions.

6.7 Projects.

Before Module 7 Vibration and Heat.

B7.1 Vibrating Strings.

B7.2 The Heat Equation.

B7.3 The Schrodinger Equation.

7 Introduction to Partial Differential Equations.

7.1 Separation of Variables.

7.2 Fourier Polynomials and Fourier Series.

7.3 Fourier Series and Separation of Variables.

7.4 Bessel and Legendre Equations.

8 Series Solutions of Differential Equations.

8.1 Expansions About Ordinary Points.

8.2 Legendre Polynomials.

8.3 Expansions about Singular Points.

8.4 Bessel Functions.

8.5 Reduction of Order.

8.6 Projects.

9 Systems of Linear Differential Equations.

9.1 Existence and Uniqueness of Solutions.

9.2 Matrix Algebra.

9.3 Diagonalizable Systems with Constant Coefficients.

9.4 Non-Diagonalizable Systems with Constant Coefficients.

9.5 Qualitative Analysis.

9.6 Variation of Parameters.

9.7 Outlook on the Theory: Matrix Exponentials and the Jordan Normal Form.

A Background.

B Tables.

C Hints and Solutions for Selected Problems.

D Activities.

Bibliography.

Index.