1550078481-Ordinary_Differential_Equations__Roberts_

Applications of the Initial Value Problem y' = f(x, y); y(c) = d 129

3.2 Learning Theory Models

Psychologists have studied the process of learning extensively. We will now

derive a few simple models of the memorization process. Let A be the total

amount of material to be memorized. Psychologists refer to the amount of

material memorized at time t as the attainment. Let y(t) denote the attain-

ment. In the simplest model of memorization, it is assumed that the rate of

change of attainment is proportional to the amount of material that remains

to be memorized. Thus, in mathematical symbolism

y'(t) = k(A -y(t))

where k > 0 is the constant of proportionality which indicates the natural

learning ability of the particular subject (person or perhaps animal). We

assume when the memorization process begins at t = 0, the subj ect has not

memorized any material. So, y(O) = 0. Hence, for the simplest memorization

model, we need to solve the initial value problem

(1) y' = k(A - y); y(O) = 0.

EXAMPLE 1 A Simple Learning Theory Model

Solve the initial value problem (1) numerically and graph the "learning

curve" (the solution to the IVP (1)) for a subj ect whose natural learning

ability is k = .06 items/minute, if the number of things to be memorized is

A= 50.

SOLUTION

We ran SOLVEIVP by setting f(x, y) = .06(50 - y) and setting the left

endpoint of the interval of integration a = 0. We needed a reasonable value

for the right endpoint of the interval of integration b. Since 0 :::; y(t) :::; 50 and
y' (t) = .06(50 -y(t)) = 3 - .06y(t), y' (t) satisfies the inequality 0 :::; y' (t) :::; 3.
Dividing the total number of items to be memorized, A, by the conservative

estimate of the rate of learning, y' = 1, we decided to set b = A/1 = 50.

At the start of the experiment the subj ect has learned nothing, so the initial
condition we used was y(O) = 0. A graph of the solution is shown in Figure 3.8.
Observe that y(t) increases rapidly fort near zero and limt_,= y(t) = 50 = A.
Thus, the largest amount of material is memorized at the beginning of the
learning experience and in a relatively short period of time. As t increases and
maximum attainment, A, is approached, the rate of attainment, y', becomes
small. This phenomenon is called the "law of diminishing returns." Displaying
our solution values on the monitor, we saw that 25 items were memorized in