What are you going to learn?

Content

Model with first-order differential equations (DE) and identify initial value problems. Solve scalar differential equations, homogeneous and non-homogeneous, using methods including separation of variables, integrating factors, eigenvalues, LaPlace transformations. Model with systems of first-order DEs and higher-order DEs. Solve systems of linear differential equations using matrices and eigenvalues.

Chapter 1. Introduction

• Terminology

• Classification of Differential Equations

• Initial Value Problem. Cauchy’s Theorem of y’ = f(x, y)

• Mathematical Models and Direction of Fields

• Integrating Factors

• Separation of Variables

• Exact Equations

• First Order Linear Equations

• Equations with Homogeneous Coefficients

• Solution by Substitutions

• Bernoulli’s DE

• Computer Solutions

Chapter 2. First Order Ordinary Differential Equations (ODE)

Chapter 3. Second Order Ordinary Differential Equations

• The General Second Order Linear Homogeneous Equation

• Linear Independence

• Existence and Uniqness Theorem

• The Wronskian

• General Solution of a Homogeneous Equation

• General Solution of a Nonhomogeneous Equation

• Differential Operators

Chapter 4. Higher Order Constant Coefficients Linear Equations

• Power Series

• Series Solutions Near an Ordinary Point

• Euler Equations, Regular Singular Points

• Series Solutions Near an Regular Singular Point

• Bessel’s Equation

Chapter 5. Series Solutions of 2d Order LDE

• Power Series

• Series Solutions Near an Ordinary Point

• Euler Equations, Regular Singular Points

• Series Solutions Near an Regular Singular Point

• Bessel’s Equation

Chapter 6. Laplace Transform

• Definition

• Solution of Initial Value Probems

• Step Fucntions

• Impulse Functions

• The Convolution Integral

Chapter 7. Systems of First Order Differential Equations

• Review of Matrices

• Systems of Linear Equations. Linear Independence

• Eigenvectors, Eigenvalues

• Basic Theory

• Homogeneous Linear Systems with Constant Coefficients

• Nonhomogeneous Linear Systems

Chapter 8. Numerical Methods

• The Euler’s Method

• Runge-Kutta Method

• Multi Step Methods

• Systems of First order Equations

Bibliography

1. Boyce, W. E., DiPrima, R. Elementary Differential Equations and Boundary Problems. 8th ed., Wiley & Sons Inc., New York.

2. Zill, D. A First Course in Differential Equations with Modeling Applications. 10th ed., Brooks/Cole, London, 1997.

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