Title

Introduction to Differential Equations
Author

 Ed Packel
Education level

College
Objectives

To appreciate how differential equations arise from modeling real-world processes and how they can be used in real-world applications

To understand what differential equations are, what it means to be a solution of one, and how solutions should be interpreted

To learn how to solve certain differential equations analytically

To learn how to investigate the nature of differential equations qualitatively and geometrically

To understand the need for and methods of numerical solution of diffential equations

To make wise use of the computer in the study of differential equations

To appreciate the value of working in teams and to learn to do so effectively
Materials

Differential Equations by Blanchard, Devaney, and Hall

VisualDSolve (Mathematica Differential Equations Package)by Schwalbe and Wagon
Description

This is a sophomore level course having 3 terms of calculus as prerequisite. It develops concepts and techniques relating to ordinary differential equations, including d.e. models, analytic solutions, qualitative analysis, and numerical solutions.

The "courseware" in this case consists of about 20 Mathematica notebooks that were used for presentations in the course. Students are given hard copies of the notebooks and they are made available to them on our server. All topics are not covered in the notebooks, just appropriate ones for Mathematica in most cases. Students or institutions would need to acquire VisualDSolve to make fully effective use of the notebooks.

Topics:

Introductory Overview: The course, differential equation models, examples

Solutions of First Order Differential Equations: Analytic (separation of variables), Geometric/Qualitative (slope fields), Numerical (Euler method); interpretation of results

Linear Differential Equations: Analytic solution by integrating factor, change of variables, examples

Differential Equations as Dynamical Systems: Equilibrium points, phase lines, and bifurcations

First Order Systems: Predator-prey models, autonomous systems, geometry and qualitative analysis

Higher Order Systems: Conversion to systems, Harmonic Oscillator and other physical examples

Linear Systems: Matrix notation, linearity principle, "straight line" solutions

Solving Linear Systems: eigenvalues/eigenvectors, complex and degenerate cases, classification

Nonlinear Systems: Equilibrium analysis, bifurcations, sensitivity, and chaos
Subject

 Mathematics > Calculus and Analysis > Differential Equations