Mathematical Methods (10/24.539)
I. First Order Differential Equations
Introduction
The goal of this course is to provide the student a solid foundation in engineering mathematics and to give him or her a set of practical tools that can be used in solving a variety of engineering design and analysis problems. The course is geared towards the senior or first-year graduate level and it reviews, enhances, extends, and integrates much of the engineering mathematics that is treated in a variety of undergraduate math and engineering science courses. The course is fast paced and it emphasizes both analytical procedures and practical numerical applications.
Since most physical systems can be described in mathematical terms via differential equations (using physical conservation laws), much of the emphasis of this course deals either directly or indirectly with the development and solution of both ordinary and partial differential equations. The topics covered gradually build a mathematical framework that gives the student the tools needed to identify and solve a variety of differential equations common to many engineering disciplines.
In building a foundation, one should always start with a strong base of familiar topics. With this in mind, this first section of notes briefly reviews the basic concepts and strategies (both analytical and numerical) for solving
First Order Ordinary Differential Equations (ODEs). The most common and most useful general solution techniques are first reviewed and then several examples illustrate how to apply the various methods. This presentation model, which includes an overview of the solution algorithms (with minimal theoretical background) and specific example applications, is followed throughout these course notes.The material here and in the next few sections is typically treated in some detail in the standard undergraduate Differential Equations course taken by all engineering students. As such, these initial sections will be reviewed rather quickly – with the goal of making sure that everyone has a common base for more advanced material.
In this first section, our summary of analytical and numerical solution schemes for first order ODEs is broken into several subsections, as follows:
using Method IV
Example 1.2 - Solve 
using Method II
Example 1.3 - Solve same equation as in Example 1.2 using Method VI
Example 1.4 - Solve 
using Method VI
Example 1.5 - Chain Sliding Off Frictionless Table
Example 1.6 - Orthogonal Trajectories
The reader should also see Appendix I for additional modeling and simulation examples involving first order ODEs.
|
|
 |
 |
10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated August 1998).
Return to Online Courses
