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 $A(t)$.)

Neetesh Kumar | November 08, 2024

Differential Equation Homework Help

This is the solution to Math 2A, section 13Z, Fall 2023 | WebAssign
Math002ACh1Sec03 (Homework) Question - 10
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Step-by-Step-Solution:

Let:

• $M$ = Total amount of the subject to be memorized.
• $A(t)$ = Amount memorized at time $t$.
• $(M - A(t))$ = Amount left to be memorized.

Step 1: Define the Rates

The rate at which material is memorized is proportional to the amount left to be memorized:

$\text{Rate of memorization} = k (M - A(t))$

Where $k > 0$ is the constant of proportionality.

Step 2: Write the Differential Equation

The change in the amount memorized over time, $\frac{dA}{dt}$, can be expressed as:

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

Conclusion

The differential equation governing the amount $A(t)$ of the subject memorized is:

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

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