3.4 Slope Fields and Euler's Method

Summary

From the introductory section, slope fields are revisited. Out of them is developed a powerful but simple procedure called Euler's Method for generating an approximate solution curve for an IVP. Ways are considered to make the Euler's Method approximation more accurate.

By the end of your studying, you should know:

How to interpret a slope field.
How to sketch a solution curve for an IVP on a slope field.
Euler's Method, both the formulas and their geometric meaning.
How to generate an Euler's Method approximation by hand.
How to generate an Euler's Method approximation with an applet.

On-screen applet instructions: The applet allows the user to generate an Euler's Method approximation to the solution of the given IVP. The buttons are fairly self-explanatory. What is the effect of changing the step size?

Examples

An investor has a savings account that pays 3.5% interest. The investor opens the account with $500 and makes an additional deposit of $500 at the end of each year. Assume the function S(t), which gives the amount in the savings account after t years, satisfies the differential equation

Use Euler's method to estimate the amount in the account at the end of 5, 10, and 25 years, using a step size of 0.1.

Let y be a function of x that satisfies the differential equation

Use Euler's method with step sizes .01, .001, .0001 to estimate the value of y when x = .99. Do these estimates seem reliable?

Assume that the point (–1, .001) is on the graph of a function that satisfies the differential equation

Estimate the value of y when x = 1 using Euler's method.