1550078481-Ordinary_Differential_Equations__Roberts_

130 Ordinary Differential Equations

the first 11.6 minutes and a total of 37 items were memorized in 22.5 minutes.
So it took almost as long to memorize items 26 through 37 (12 items) as it
did to memorize the first 25 items. Thus, later in the learning experience it
takes a longer period of time to learn the same number of items.

50

40

30

y(x)

20

10

0

(^0 10 20) x 30 40 50
Figure 3.8 Graph of the "Learning Curve": y' = 3 - .06y; y(O) = 0
EXERCISES 3.2

l. When the learning process is lengthy, k is not a constant but a func-

tion which decreases with time because the subj ect becomes fatigued or

distracted. Suppose k(t) = k 0 /(1 + .05t) where ko is a constant. Solve

numerically and graph the learning curve for the initial value problem

(2)

, ko(A - y)

y = 1 + .05t ; y(O) =^0

with ko = .0 6 items/minute and A = 50 items on the interval [O, 50].

When has the subject memorized 25 items? 37 items? Compare these

results with the corresponding answers for example l.

2. When a subj ect learns, he also forgets. In order to include the process
   of forgetting in our model of the memorization process, we assume the
   rate at which a subject forgets is proportional to the amount already
   learned. Thus, we wish to solve the following initial value problem

(3) y' = k(t)(A - y) - By; y(O) = 0