# Differential Equations

Spring 2021

## Extra stuff

## Study Tools

- Table of Integrals Anton
- Paul's Differential Equations Notes (on-line)
- Paul's Notes also has Notes/Review of Calculus I and Calculus II.
- Paul's Cheat Sheets - from algebra to Calculus
- If you need further help there is a video library of math tutorials available at the Khan Academy (free). Please let me know if any of these are helpful to any of you.

## Graphing Tools

The purpose of this section is to provide Math 240 students with a little extra help
with some of the graphing tools that we use for the qualitative
analysis of ODEs. Within the material below, you will find
short discussions about Slope Fields and Phase Portraits.
There are also a few links to some *mathy* websites that might
help you develop a better understanding of these important
topics.

Aside from the discussions, the following content provides students with some getting-started training and the links to use the web-based graphing tools that we hope you find useful and convenient. Plotting slope fields and phase portraits can get a little cumbersome at times. So, once you have the basic concepts down, feel free to use these tools while you are doing your homework or studying. Hopefully, it will save you some time. With all that time you save, you can spend a good portion of it asking yourself: "What does this graph mean?" or "What information can be gathered from this graph?"

**Note:** Using these graphics tools requires a good amount of organization
of your computer's desktop windows. Have a little patience the first couple of times, and experiment with
moving windows around so that you can find them all. When all else fails MINIMIZE!

**Graphing Functions**- Since most of you are Windows users, you may wish to try Winplot. This is a great -FREE- math ploting utility. (sorry, no Linux equivalent now).

**Slope Fields**- Dfield Introduction, followed by link below.
- dfield and pplane Tools by John C. Polking at Rice University (did you look at the syntax rules in the link above?)
- Qualitative Techniques: Slope Fields

**Phase Planes**