## 3.1 Modeling with Differential Equations

### Summary

This section introduces the issues to be studied in Chapter 3: Modeling with Differential Equations. Given certain differential equations, both analytical and numerical (approximate) methods will be discussed for producing solutions. Moreover, the more general notion of obtaining a function f from f' will be pursued.

By the end of your studying, you should know:

• How to solve a differential equation by inspection (guess-and-check).
• What a slope field is.
• How to plot a slope field both by hand and with an applet.
• The method of separation of variables,and how and where to apply it.

On-screen applet instructions: The applet draws a slope field for an equation y' = f(x,y). It allows several choices of the function, and allows changing the resolution of the grid. Drag the mouse to draw a solution curve through the point P. Click the mouse (instead of drag) to set a new initial point P. Click here for more details.

### Examples

Match three equations with the differential equations they satisfy.

Match four differential equations with their slope fields.

Solve the following differential equation by separation of variables.

Use the initial condition

to solve for the unique solution.

Slope Field

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 3.1 Modeling with Differential Equations (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

### Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

DEs and Slope Fields
Tool
Mathlet

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel