TI-92 Help -- Numerical Solutions of Differential Equations

x' = f1(t, x, y) and y' = f2(t, x, y)

The two TI-92 programs rk2x and rk2y listed below can be used to find numerical approximations for initial value problems of the form

on the interval [0, T]

Skip ahead if these programs are already on your TI-92. If not, enter them in the usual way by typing them in or downloading them using the TI-Graph Link.

Missing Program

There have been numerous problems downloading programs for TI graphing calculators using the method above. If that method works, it is very clean. Sometimes, however, it doesn't work. In that case we can fall back on an older method, uuencoding, that is less user-friendly but more reliable. Click here for more information about this method. Then click here for a uuencoded (text file).

These programs use a method called the Fourth Order Runge-Kutta Method that is similar in spirit to Euler's Method but is more precise.

The first step in using these programs is to define the two functions f1(t, x, y) and f2(t, x, y) that appear on the right hand side of the two differential equations.

The screen at the right shows how this is done for the example
x' = y y' = -3 x
Notice that the names of the functions must be f1 and f2 and that the variables t, x, and y must appear in that order.
The next step is to decide the step size and number of steps to be used. If the number of steps is N then the step size is
T h = --- N

Missing TI-92 screen

For our example we will examine the initial value problem
x' = y, x(0) = 0 y' = -3 x, y(0) = 1 on [0, 10]
using 200 steps, so the step size will be 0.05. Store the step size in h as shown in the screen on the right.

Next we want to set up the WINDOW screen to graph the solution. The screen at the right shows the set-up for our example. The TI-92 graphics mode must set to Sequence. You may want to look at the help module for Graphing a Sequence.
Notice that

nmin = 0

nmax = N -- the number of steps.

xmin = 0

xmax = N -- the number of steps

Missing TI-92 screen

The next step is to set up the Y= screen as shown in the screen at the right.
The sequences u1(n), u2(n), and u3(n) must be set up exactly as shown. The sequence u1 gives the time at step n and the sequences u2(n) and u3(n) give the value of x and y, respectively, at step n. Notice that the graph for u1(n) is turned off and the graphs for u2(n) and u3(n) are turned on. The F4 key is used to turn the graph for a particular sequence on or off. Use the cursor pad to highlight the sequence in question and press F4 to change it from on to off or vice versa.
The initial value x(0) is the initial value -- ui2 -- of the sequence u2(n). The initial value y(0) is the initial value -- ui3 of the sequence u3.
Missing TI-92 screen

Now press green-diamond GRAPH to see the solution of this initial value problem as shown in the screen at the right.

You can also examine the solution numerically by using the TI-92 table feature. Set up the TblSet screen as shown in screen at the right. Missing TI-92 screen

Then turn the u1(n) sequence on in the Y= screen and press green-diamond TABLE to see a screen like the one at the right.

The screen at the right shows how this is done for the example x' = y y' = -3 x Notice that the names of the functions must be f1 and f2 and that the variables t, x, and y must appear in that order. The next step is to decide the step size and number of steps to be used. If the number of steps is N then the step size is T h = --- N
For our example we will examine the initial value problem x' = y, x(0) = 0 y' = -3 x, y(0) = 1 on [0, 10] using 200 steps, so the step size will be 0.05. Store the step size in h as shown in the screen on the right.
Next we want to set up the WINDOW screen to graph the solution. The screen at the right shows the set-up for our example. The TI-92 graphics mode must set to Sequence. You may want to look at the help module for Graphing a Sequence. Notice that nmin = 0 nmax = N -- the number of steps. xmin = 0 xmax = N -- the number of steps
The next step is to set up the Y= screen as shown in the screen at the right. The sequences u1(n), u2(n), and u3(n) must be set up exactly as shown. The sequence u1 gives the time at step n and the sequences u2(n) and u3(n) give the value of x and y, respectively, at step n. Notice that the graph for u1(n) is turned off and the graphs for u2(n) and u3(n) are turned on. The F4 key is used to turn the graph for a particular sequence on or off. Use the cursor pad to highlight the sequence in question and press F4 to change it from on to off or vice versa. The initial value x(0) is the initial value -- ui2 -- of the sequence u2(n). The initial value y(0) is the initial value -- ui3 of the sequence u3.
Now press green-diamond GRAPH to see the solution of this initial value problem as shown in the screen at the right.
You can also examine the solution numerically by using the TI-92 table feature. Set up the TblSet screen as shown in screen at the right.
Then turn the u1(n) sequence on in the Y= screen and press green-diamond TABLE to see a screen like the one at the right.