Week 2 (September 6 - September 10)

In this course we will describe procedures for obtaining symbolic solutions for two classes of first-order differential equations. This week we will consider separable equations and next week linear first-order equations.

The second piece of business this week will be to discuss numerical methods for approximating the solutions of first-order differential equations with initial conditions -- i.e., initial value problems. Because we will make heavy use of computer generated numerical solutions in this course, this is a much more important topic for us than it is in a more old-fashioned course. Thus, we will jump to a brief look at the main ideas behind numerical solutions in Chapter 8.

Also, we will work through the module Numerical Solutions of Differential Equations. Part 1 of this module describes Euler's Method -- the simplest of the numerical methods. This is just reading and can be read outside of class in parallel with Section 8.1 in the text. Part 2 of the module gives you practice with Euler's Method and several improved methods. The improved methods are described in Sections 8.3 and 8.4 of the text. Part 3 of the module looks at the logistic equation -- an example of a separable equation. Part 4 looks at a differential equation with many similarities to the logistic equation, but not one you want to try to solve symbolically.

Reading

• Section 2.3: A discussion of separable equations

• Section 8.1: Read to the top of page 419.

• Section 8.3: Read to the top of page 432.

• Section 8.4: Look over this to see what goes into the standard fourth-order Runge-Kutta method.