1. Reading Between the Lines!
Slope Fields and Euler’s Method

2. Slope Fields
A first order differential equation can be written as a condition in the form in which the slope of the function can be calculated as a function of x and y

3. What function or family of functions produces a slope field like this?
This sometimes enables us to determine the correct form of y(x). For example:
What function or family of functions produces a slope field like this?

4. What function or family of functions produces a slope field like this?

5. Your turn…
Sketch slope fields for the following and try to guess the mathematical function or relation that these represent: a b c

6. Euler’s Method…
Slope fields also suggest an algebraic approximation scheme that we could use to numerically solve a differential equation

7. Example…
Compare an Euler’s solution of with an exact solution. Use h = 0.1 and plot the solution between x = 0 and x = 10. Use Excel.

8. Some Worked Examples…
Look at 11.3 # 8,9 in Excel

9. Working With DEs and Separation of Variables
Method is very useful for many cases in which variables can be isolated from one-another
Requires that isolated (separated) terms are integrable
If separation doesn’t work can always fall back on Euler’s Method
Examples from 11.4; 1, 15, 27