This page discusses desolve, Euler's method, and plotting approximate solutions using direction fields.
The calculator has a built-in utility that finds exact solutions to many differential equations.
The command is called desolve; it's option 3 on the Calc menu, F3.
The syntax is desolve( y' = 2x + 3y, x, y). The prime symbol is 2nd B. After the differential equation, you enter the names of the input variable and the output variable, in that order. They don't have to be x and y. The spaces in the command are only there to make it easier to read; you don't need them when you enter the command.
If you want the calculator to solve an initial value problem, the syntax is desolve( y' = 2x + 3y and y(0) = 4, x, y). You do need spaces around the and.
The calculator renders the arbitrary constants in the solution with symbols like @3. You should think of this as C3, the third arbitrary constant. (The calculator restarts its numbering with @1 every time you enter NewProb.) If you want to type the @, so that you can substitute a value for it or solve for it, for instance, the key is 2nd R.
To solve a second order differential equation, enter y'' by typing 2nd B twice. (Don't use 2nd L to get the quotation mark ".) The calculator won't solve third order differential equations.
Be sure to enter the product xy as x*y.
There is no built-in way to run Euler's Method on the calculator, but it's easy to create one.
Suppose we want to use Euler's method to find an approximate solution to the differential equation y' = 1 - xy. We'll assume y(0) = 1, and that we want to use a step size of 0.1 to approximate y(1).
Go to the HOME screen and enter NewProb (F6,2).
Enter 1-x*y -> f(x,y) (The -> is meant to represent the STORE key, next to the spacebar.)
Enter 0.1 -> d : 0 -> x : 1 -> y (The colons allow you to enter several commands on one line. Although there are spaces in the line above to help you read it, you don't need any spaces when you type it on the command line.)
Enter y + f(x,y)*d ->y : x + d -> x : { x, y } (You do want the colons here, because you want the x and y values displayed, but not the calculations that produce them.)
Before you do anything more, use F1,2, Save Copy As?, to save a copy of this text file under the name Euler.
Every time you hit ENTER while the text y + f(x,y)*d ->y : x + d -> x : { x, y } remains on the command line, the calculator will compute one more step in Euler's Method and display the x- and y-coordinates of a point on the (approximate) graph of the solution to the differential equation. Keep hitting ENTER until the output is {1, something}. The "something" is your estimate of y(1).
To repeat the process with a different differential equation or a different initial condition or a different step size, use the Text Editor. Use the APPS key next to the cursor control, choose option 8: Text Editor, and select option 2: Open? You'll get a dialog box in which you can select Euler. There should be four lines, each starting with a C:, which identifies the lines as commands.
First, edit the commands, changing the differential equation (you enter the formula for y' at the beginning of the second line, so that it's stored in f(x,y)), or changing the step size (stored in d in the third line) or initial conditions (the values stored in x and y in the third line). Leave the first and fourth lines alone.
Next, put the cursor somewhere on the first line and execute by hitting F4 four times.
Now, go to the HOME screen. The calculator will have started on Euler's Method. If you wanted to estimate y(3), say, keep hitting ENTER until the first number in the output is 3. The second number will be your estimate of y(3).
The TI-92+ will plot slope fields (also called direction fields) for first order differential equations and draw graphs of solutions to those differential equations on the same axes.
First, use the MODE key to change the GRAPH MODE to DIFF EQUATIONS.
Go to the Y= screen and use F1,8 to clear out any old equations.
Enter a differential equation. The input variable is always t. The output variable is always y1. (You can use y2, y3, etc., but you can only get slope fields for one differential equation at a time. You may enter an initial condition on the next line by setting yi1 equal to y1(0). If you're given a value for y1(3) instead, change the line t0=0 to t0=3 and enter y1(3) on the yi1 line. But you can also enter initial conditions interactively on the graph screen, and this works better.)
If necessary, adjust the graph format by entering F1,9. The settings you want are
Coordinates RECT Grid OFF Axes ON Leading Cursor OFF Labels OFF Solution Method RK Fields SLPFLD(These are the default settings, so you shouldn't have to change anything if you've never graphed a differential equation before.)
Go to the WINDOW screen and adjust the window parameters to show the part of the x-y plane (actually the t-y1 plane) you want. You want ncurves = 0, diftol = .001, and fldres = 20; I think these are the default settings.
Go to the GRAPH screen and see the slope field associated with the differential equation.
To have the calculator plot a solution, use F8, labeled IC for "initial condition." You may either move the cursor by hand or enter values from the keyboard for t and y1. When you hit ENTER, the calculator will draw the graph of a solution to the differential equation that passes through the point at which the cursor is located. The calculator will graph as many different solutions as you ask it to. However, the TRACE command is not active, so you can't read off coordinates of points on your solution curve the way you can with graphs of functions.
Last updated 12-6-2002, wdj