Math 4B

Course Topics.(Sections numbers from Differential Equations with Boundary Value Problems, by Zill-Cullen, Fifth Edition, Brooks/Cole).

Categories: 1. Must be covered thoroughly.

2. Must be covered, leaving details at teacher’s discretion.

3. Cover only when there is extra time.

Category Section Topic

1 1.1 Definitions and Terminology

2 1.2 Initial-Value Problems

2 1.3 Differential Equations as Mathematical Models

FIRST-ORDER DIFFERENTIAL EQUATIONS

2 2.1 Solution Curves Without the Solution

1 2.2 Separable Variables

1 2.3 Linear Equations

1 2.4 Exact Equations

1 2.5 Solutions by Substitutions

2 2.6 A Numerical Solution

MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS

2 3.1 Linear Equations

3 3.2 Nonlinear. Equations

3 3.3 Systems of Linear and Nonlinear Differential Equations

HIGHER-ORDER DIFFERENTIAL EQUATIONS

1 4.1 Preliminary Theory: Linear Equations

1 4.1.1 Initial-Value and Boundary-Value Problems

1 4.1.2 Homogeneous Equations

1 4.1.3 Nonhomogeneous Equations

1 4.2 Reduction of Order

1 4.3 Homogeneous Linear Equations with Constant Coefficients

3 4.4 Undetermined Coefficients-Superposition Approach

1 4.5 Undetermined Coefficients-Annihilator Approach

1 4.6 Variation of Parameters

1 4.7 Cauchy-Euler Equation

2 4.8 Solving Systems of Linear Equations by Elimination

3 4.9 Nonlinear Equations

MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS

1 5.1 Linear Equations: Initial-Value Problems

1 5.1.1 Spring/Mass Systems: Free Undamped Motion

1 5.1.2 Spring/Mass Systems: Free Damped Motion

1 5.1.3 Spring/Mass Systems: Driven Motion

1 5.1.4 Series Circuit Analogue

2 5.2 Linear Equations: Boundary-Value Problems

3 5.3 Nonlinear Equations

SERIES SOLUTIONS OF LINEAR EQUATIONS

1 6.1 Solutions About Ordinary Points

1 6.1.1 Review of Power Series

1 6.1.2 Power Series Solutions

1 6.2 Solutions About Singular Points

2 6.3 Two Special Equations

THE LAPLACE TRANSFORM

1 7.1 Definition of the Laplace Transform

1 7.2 Inverse Transform and Transforms of Derivatives

1 7.3 Translation Theorems

1 7.3.1 Translation on the s-Axis

1 7.3.2 Translation on the t-Axis

1 7.4 Additional Operational Properties

3 7.5 Dirac Delta Function

2 7.6 Systems of Linear Equations

SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS

Chapter 8 is in category 3

8.1 Preliminary Theory

8.2 Homogeneous Linear Systems with Constant Coefficients

8.2.1 Distinct Real Eigenvalues

8.2.2 Repeated Eigenvalues

8.2.3 Complex Eigenvalues

8.3 Variation of Parameters

8.4 Matrix Exponential

NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

2 9.1 Euler Methods and Error Analysis

2 9.2 Runge-Kutta Methods

3 9.3 Multistep Methods

3 9.4 Higher-Order Equations and Systems

3 9.5 Second-Order Boundary-Value Problems

ORTHOGONAL FUNCTIONS AND FOURIER SERIES

1 11.1 Orthogonal Functions

1 11.2 Fourier Series

1 11.3 Fourier Cosine and Sine Series

3 11.4 Sturm-Liouville Problem

3 11.5 Bessel and Legendre Series

3 11.5.1 Fourier-Bessel Series

3 11.5.2 Fourier-Legendre Series

PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY-VALUE

PROBLEMS IN - - RECTANGULAR COORDINATES

1 12.1 Separable Partial Differential Equations

2 12.2 Classical Equations and Boundary-Value Problems

1 12.3 Heat Equation

1 12.4 Wave Equation

2 12.5 Laplace's Equation

3 12.6 Nonhomogeneous Equations and Boundary Conditions

3 12.7 Orthogonal Series Expansions 551

3 12.8 Boundary-Value Problems Involving Fourier Series

Note: It is extremely important to cover the category 1 topics in chapters 11 and 12.

Without these topics students would not be able to take at Mission D. E. as an upper division course.