# Euler's method on the TI-82/83

Given a differential equation, say y' = 3x-y, here is how to make the TI-82/83 draw an Euler's method solution for that differential equation.

1. We must translate the differential equation: In place of the function (y in this case), write Y (upper case), in place of y' write Y1, and in place of the variable (x in this case), write X. So we get Y1=3X-Y.
2. Now press [Y=] and enter our differential equation from the last step on the first line:

3. Press [WINDOW] to select a suitable window or range. Here's what I've picked:

4. Press [PRGM] and select the EULER program.  Press [ENTER]once to paste prgmEULER on your text screen; press [ENTER] again to run the program.
5. Remember that Euler's method requires an initial condition or starting point for a solution curve.  Suppose, for examples, that the desired initial condition was y(1) = 4. When the program begins, you must input that initial condition as follows:
1. When the program asks "X START?", give the x-coordinate of the initial condition.  In our example y(1) = 4, X START would be 1.
2. When the program asks "Y START?", give the y-coordinate of the initial condition. In our example y(1) = 4, Y START would be 4.
6. Euler's method can run forever; you must tell the program when to stop.  If you want a solution curve from X START (1 in the above example) to, say, 5, you would give "5" for "FINAL X".
7. Finally, give the number of steps desired (so the stepsize works out to be (FINAL X - X START)/(# of steps) ).
8. Your screen will look something like this before you press [ENTER] the last time:

9. When you press [ENTER] the last time, the calculator graphs an approximate solution curve to the differential equation, for the given initial condition:

10. You can get the coordinates of the last point plotted by typing X [ENTER] [ALPHA] Y [ENTER]:

A more accurate differential equations solution grapher is available. It is similar to the better methods built in to the TI-85/86, though not quite as sophisticated. It will generally require far fewer steps for far greater accuracy.