Question

In Problem 25 assume that the rate at whichmaterial is forgotten is proportional to the amount memorized in time $\mathbf{t}\mathbf{>}\mathbf{0}$. Determine a differential equation for the amount $\mathbf{A}\left(\mathbf{t}\right)$when forgetfulness is taken into account.

Short answer

The differential equation is $\frac{dA}{dt}=c_1(M-A(t\left)\right)-c_{2}A\left(t\right)$.

Step-by-step solution

Definition of Newton’s law of cooling with the mathematical statement

Newton’s law of cooling/warming translates into the mathematical statement

$\frac{dT}{dt}\propto T-T_m\text{or}\frac{dT}{dt}=k\left(T-T_m\right)$

Where k is a constant of proportionality. In either case, cooling or warming, if $T_m$is a constant, it stands to reason that $k<0$.

Compute Differentiation

Using the fact that the rate of memorizing is proportional to the amount left to be memorized, we know that

$\frac{dA}{dt}\propto M-A\left(t\right)$

Which can be written as

$\frac{dA}{dt}=c_1(M-A(t\left)\right)$

Where $c_1$is constant, $c_1\ne 0$.

Besides, rate at which material is forgotten is proportional to the amount memorized at time $t$, which we write down mathematically as

$\frac{dA}{dt}=-c_{2}A$

Therefore, DE for $A\left(t\right)$when forgetfulness is taken into account will be $\frac{dA}{dt}=c_1(M-A(t\left)\right)-c_{2}A\left(t\right)$.