1. First Peoples Buffalo Jump State Park, near Great Falls, Montana Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2009 6.1 day 2 Euler’s Method

2. 6.1 day 2 Euler’s Method Leonhard Euler 1707 - 1783 Leonhard Euler was a Swiss mathematician who made a huge number of contributions to mathematics, almost half after he was totally blind. (When this portrait was made he had already lost most of the sight in his right eye.) Pronounced: oi-ler

3. Leonhard Euler 1707 - 1783 It was Euler who originated or popularized the following notations: (base of natural log) (function notation) (pi) (summation) (finite change) The symbol was first used in 1706 by Welsh mathematician William Jones, 1675-1749. However, the symbol was not in common use until 1748 when Euler used it in Introductio in Analysin Infinitorum.

4. There are many differential equations that can not be solved. We can still find an approximate solution. We will practice with an easy one that can be solved. Initial value:

7. Exact Solution:

8. It is more accurate if a smaller value is used for dx. This is called Euler’s Method. It gets less accurate as you move away from the initial value.

9. The TI-89 has Euler’s Method built in. Example: We will do the slopefield first: 6: DIFF EQUATIONS Graph….. Y= We use: y1 for y t for x MODE

10. 6: DIFF EQUATIONS Graph….. WINDOW t0=0 tmax=150 tstep=.2 tplot=0 xmin=0 xmax=300 xscl=10 ymin=0 ymax=150 yscl=10 ncurves=0 diftol=.001 fldres=14 not critical GRAPH We use: y1 for y t for x Y= MODE

11. t0=0 tmax=150 tstep=.2 tplot=0 xmin=0 xmax=300 xscl=10 ymin=0 ymax=150 yscl=10 ncurves=0 diftol=.001 fldres=14 WINDOW GRAPH

12. While the calculator is still displaying the graph: yi1=10 tstep =.2 If tstep is larger the graph is faster. If tstep is smaller the graph is more accurate. I Press and change Solution Method to EULER. WINDOW GRAPH Y=

13. To plot another curve with a different initial value: Either move the curser or enter the initial conditions when prompted. F8 You can also investigate the curve by using. F3 Trace

14. The book refers to an “Improved Euler’s Method”. We will not be using it, and you do not need to know it. The calculator also contains a similar but more complicated (and more accurate) formula called the Runge-Kutta method. You don’t need to know anything about it other than the fact that it is used more often in real life. This is the RK solution method on your calculator. 