Question

In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t > 0. Determine a differential equation for the amount A(t). (Assume the constant of proportionality is k > 0. Use A for A(t).)

Answer 1

Step-by-step explanation:

From the statement:

M: is total to be memorized

A(t): the amount memorized.

The key issue is translate this statement as equation "rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized"

memorizing rate is $\frac{dA(t)}{dt}$.

the amount that is left to be memorized can be expressed as the total minus the amount memorized, that is $M-A(t)$.

So we can write

$\frac{dA(t)}{dt}=k(M-A(t))$

And that would be the differential equation for A(t).