This module has two parts. In the first part we study Euler's Method, a straightforward but not very efficient method for estimating solutions of an initial value problem. In the second part we use a much more efficient TI-92 based numerical method to investigate several variations of Newton's Model of cooling. The TI-92 help module Differential Equations of the form y' = f(t, y) discusses this more efficient method.
In the remainder of this window we discuss how you can use your TI-92 to check your calculations for Euler's Method.
The first step is to define the function f(t, y) on the right hand side of the differential equation
dy
-- = f(t, y)
dt
as shown in the screen below.
Next we define the stepsize h as shown in the screen below.
Next set the TI-92 graphics mode to SEQUENCE in the MODE screen.
We will use two sequences, u1(n) and u2(n) to keep track of t(i) and w(i), respectively. Notice that on the TI-92 we use n instead of i. The following screen shows how the TI-92 Y= screen can be used to define these sequences.
The initial value, ui1, of the sequence u1 is the left endpoint, a, of the time interval [a, b]. The initial value, ui2, of the sequence u2 is the initial value of the variable y.
Next set-up the TblSet screen as shown in the screen below.
and then press green-diamond-TABLE to see the results of using Euler's method as shown in the screen below.
You can examine the results graphically by turning off the u1 plot using F4 in the Y= screen as shown in the screen below.
Then setting the WINDOW screen as shown in the screen below
and then pressing green-diamond GRAPH to see the screen below.
