TI-92 Help -- Numerical Solutions of Differential Equations

y' = f(t, y)

The TI-92 program below -- rk1 -- can be used to find numerical approximations for solutions of initial value problems of the form

on the interval [0, T].

Skip ahead if this program is already in your TI-92. If not, enter it in the usual way by typing it in or by downloading it using the TI-Graph Link.

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).

This program uses a method called the Fourth Order Runge-Kutta Method that is similar in spirit to Euler's Method but is more precise. For differential equations of the form

on the interval [0, T]. Euler's Method is the same as the left edge method for estimating integrals and the Fourth Order Runge-Kutta Method is the same as Simpson's Rule.

To use rk1 the first step is to define the function
f(t, y)
appearing on the right side of the differential equation. The screen at the right shows how the right hand side of the differential equation
y' = (sin (2 Pi t) - y)
is entered.
Missing TI-92 screen

Notice that the name of the function must be rk1f and the time variable t must be first.
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
y' = (sin (2 Pi t) - y) y(0) = 2 on [0, 6]
using 120 steps, so the step size will be 0.05. Store the step size in h as shown in the screen at the right.
Missing TI-92 screen

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) and u2(n) must be set up exactly as shown. The sequence u1 gives the time at step n and the sequence u2(n) gives the value of y at step n. Notice that the graph for u1(n) is turned off and the graph for u2(n) is 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 y(0) is the initial value -- ui2 -- of the sequence u2(n).
The sequence u3(n) in the screen above gives us an extra bit of information for this particular differential equation. It graphs the function sin(2 Pi t). For this particular initial value problem it is interesting to compare this function with the solution of the initial value problem.
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.

To use rk1 the first step is to define the function f(t, y) appearing on the right side of the differential equation. The screen at the right shows how the right hand side of the differential equation y' = (sin (2 Pi t) - y) is entered.
Notice that the name of the function must be rk1f and the time variable t must be first. 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 y' = (sin (2 Pi t) - y) y(0) = 2 on [0, 6] using 120 steps, so the step size will be 0.05. Store the step size in h as shown in the screen at 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) and u2(n) must be set up exactly as shown. The sequence u1 gives the time at step n and the sequence u2(n) gives the value of y at step n. Notice that the graph for u1(n) is turned off and the graph for u2(n) is 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 y(0) is the initial value -- ui2 -- of the sequence u2(n). The sequence u3(n) in the screen above gives us an extra bit of information for this particular differential equation. It graphs the function sin(2 Pi t). For this particular initial value problem it is interesting to compare this function with the solution of the initial value problem.
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.