in the theory of learning the rate at which the subject is memorized is assumed to be directly proportional to the amount left to be memorized. also the rate at which the subject is forgotten is directly proportional to the amount memorized in their time t. suppose M denotes the total amount to be memorized and A(t) is the amount memorized at any time t. Determine the differential equation of A(t)
dA/dt = k(M-A) - cA
dA/dt = k M - kA - c A
but k - c is a constant call it r
dA/dt = kM- rA
dA/dt +rA = k M = constant for this case of M
To determine the differential equation for A(t), let's break down the given information:
1. "The rate at which the subject is memorized is assumed to be directly proportional to the amount left to be memorized."
This implies that the rate of change of A(t) with respect to time (dA/dt) is directly proportional to the amount left to be memorized (M - A(t)).
2. "The rate at which the subject is forgotten is directly proportional to the amount memorized in their time t."
This implies that the rate of change of A(t) with respect to time (dA/dt) is directly proportional to the amount memorized (A(t)).
Combining these two pieces of information, we can write the differential equation for A(t) as:
dA/dt = k1 * (M - A(t)) - k2 * A(t)
Where:
- dA/dt is the rate of change of A(t) with respect to time.
- k1 is the constant of proportionality for the rate of learning.
- k2 is the constant of proportionality for the rate of forgetting.
This differential equation represents the dynamic relationship between the rate of change of the amount memorized and the amount left to be memorized, as well as the rate of forgetting.
To determine the differential equation of A(t), let's use the given information and assumptions.
According to the problem statement, the rate at which the subject is memorized is directly proportional to the amount left to be memorized. We can represent this relationship with the following proportionality constant, k1:
dA/dt = k1 * (M - A(t))
This equation represents the rate of change of the amount memorized, dA/dt, which is equal to the proportionality constant k1 times the amount left to be memorized (M - A(t)).
Similarly, the rate at which the subject is forgotten is directly proportional to the amount memorized. This relationship can be represented by another proportionality constant, k2:
dA/dt = -k2 * A(t)
This equation represents the rate of change of the amount memorized, dA/dt, which is equal to the negative proportionality constant k2 times the amount memorized, A(t).
Since the rate of change of the amount memorized can be affected by both the memorization and forgetting processes, we can combine these two equations:
dA/dt = k1 * (M - A(t)) - k2 * A(t)
This is the differential equation for A(t) according to the given assumptions and information:
dA/dt = k1 * M - (k1 + k2) * A(t)