## Euler's Method on the Texas Instruments TI-85 Calculator

This program applies Euler's method to the solution of the differential equation y'=f(x,y) on the x-interval [A,B]. The program prompts for input of A, B, an initial value y = y(A) and the number N of steps. It displays the resulting approximate value for y(B).

General: To type an upper case letter, press ALPHA key, followed by the letter. To type a lower case letter, press 2nd key, then ALPHA key, followed by the letter. In navigating menus, use EXIT to move back.

To enter the program commands, press PRGM key, then F2 key (which selects the EDIT box), then the program name (e.g. EULER) followed by the ENTER key. The calculator will go to a new line and display the colon : which must appear at the beginning of each command. After typing each command, press the ENTER key. Calculator will go to a new line and display the colon. The function f is stored as y1 (see below).

```Program                   Explanation and Keying instructions

:Prompt A,B,y,N              Program will ask you to key in A (initial
x-value), B (final x-value), y = y(A) the
initial y-value and N (the number of equal
subdivisions to be used). To get Prompt,
press F3 for the I/O menu, then F2.
:(B-A)/N -> H                Calculate the step-size H. -> is one
character, obtained by pressing STO.
:A -> x                      Initialize x (use the x-VAR key).
:1 -> I                      Initialize the counter I
:Lbl P                       Label this spot in the program as P.
To get Lbl, press F4 (CTL menu), MORE
(to scroll the CTL menu) and F4.
:y + y1*H -> y               The new y value is computed from the
:x + H -> x                  current x and y values. Then x is
updated. For y (lower case!) use 2nd
ALPHA; for *, press the multiplication
key, which is to the right of the 9 key.
:IS>(I,N)                    IS> increments I by 1 and tests if the
new I is larger than N. If so, the program
skips the next step(so the program displays
the current value of y and terminates).
Otherwise it cycles back to location P.
For IS>(, press F4 key (chooses CTL
menu), then twice the MORE key, then F1.
Note: the comma is the key  to the left
of the 4 key.
:Goto P                      Goto from the CTL menu.
:Disp y

```
To finish keying the program hit EXIT. To enter the function f, press GRAPH key, then F1 key, then key in the expression for f you want to sum (e.g. y). Use x-VAR for x. Then EXIT, EXIT to get back to the home screen.
Now press PRGM, then F1, then the F-key under the word EULER in the menu, then the ENTER key. Enter numbers at the question mark (?) prompts.

Check Enter y as y1, so the differential equation in question is y'= y, with solution y = C exp(x). Run the program with A=0, B=1, y=1, N=10. The result should be 2.5937424601.

Exercise: Show that if f(x,y) (the expression entered as y1) is a function of x alone, then the number this program gives with initial value y=0 is the left-hand sum approximation to the integral of f from A to B with N equal subdivisions.

Anthony Phillips
Math Dept SUNY Stony Brook
tony@math.sunysb.edu
September 7 1997