For this module you will need the following TI-92 programs. As usual you can obtain these programs by transferring them via the linking cable from another TI-92, from this computer using the TI-Graph Link92 program and cable or, as a last resort, by typing them in. Click here to skip ahead if you have the programs.
onestep
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).
ndsolve
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).
foobar
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 will enable you to find numerical estimates for solutions to systems of differential equations like the ones in this module. We illustrate how they are used by looking at the system below from this module.
I'(t) = c (B(t) - I(t))
B'(t) = k(E(t) - B(t)) + k(I(t) - B(t))
In this system E(t) is the exterior temperature and will be given by the function
E(t) = sin (pi t / 12)
The function B(t) represents the temperature of a brick and the function I(t) represents the interior temperature within an adobe house.
Our first step is to define the exterior temperature as shown in the TI-92 screen below.
The programs we are using are fairly powerful and enable us to use fairly big steps with reasonable accuracy. The variable h will be used for the step size. We will use h = 0.50 hours. We will look at a solution over a period of three days or 72 hours. Since each step will be 0.50 hours we need 144 steps.
We will work with the following values of the constants c and k.
k = 0.50
c = 0.50
and the following initial conditions
I(0) = 3
B(0) = 2
The programs above are very general and can be used in many, very different situations. For example, you can use the same programs to work with 2 or more subbrick models. The secret behind this versatility is the use of vectors. This also makes the programs a little bit more difficult to use and to understand. This a common trade-off -- more power often comes at the expense of more sophistication and difficulty.
In our example we have two differential equations -- one for the function B(t) and one for the function I(t). We use a three-dimensional vector to keep track of the two functions B(t) and I(t) and of the time t. We write a typical vector using the notation (t, B, I). For example, the initial condition for our example is (0, 2, 3) because the initial time is t = 0, the initial temperature of the brick is 2 and the initial temperature of the interior is 3.
The screen below illustrates how we specify all the information above except the differential equations.
The last line in the screen above specifies the initial conditions and the number of steps. Notice the syntax -- the number of steps is given first and followed by the initial conditions written as a vector. The program ndsolve determines the number of differential equations by looking at the length of this vector. This line will take some time to evaluate.
The differential equations are specified in the function foobar that you have already loaded into your TI-92. You will need to change this function to work with other systems of differential equations. The function foobar is shown in the screen below using the Program Editor.
This program defines three differential equations that describe the derivatives of the variables t, brick, and interior. The first differential equation is easy. The derivative of t is just 1. This is represented by the line
in the program foobar above.
The second differential equation
is represented by the line
Notice that instead of the variable t on the right hand side of the differential equation we use v[1]; instead of the function B(t) we use v[2]; and instead of the function I(t) we use v[3].
The third differential equation
is represented by the line
The final line of the program foobar (not counting the EndFunc line) is the line
This line assembles the preceding calculations for the three derivatives into one vector and returns this vector as output to be used by the program ndsolve.
After the program ndsolve in the second TI-92 screen above has finally completed its job you can graph the results as follows. First, use the SEQUENCE graph mode and make sure that the AXES are set to TIME.
Then set up the Y= screen as shown in the TI-92 screen below.
and the WINDOW screen as shown in the TI-92 screen below.
Finally, press GRAPH to see the following graph.
You may want to change the ymin setting in the window screen to -1 to get a larger graph. You may also want to change the style of one of the graphs in the Y= screen to make it easier to distinguish the graphs from each other.