This program applies Euler's method to the solution of the differential equation y'=

*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).

To finish keying the program hit EXIT. To enter the functionProgram 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

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.

Math Dept SUNY Stony Brook

tony@math.sunysb.edu

September 7 1997