Problem 42 - Advanced Engineering Mathematics

Question

When forgetfulness is taken into account, the rate of memorization of a subject is given by $$ \frac{d A}{d t}=k_{1}(M-A)-k_{2} A $$ where \(k_{1}>0, k_{2}>0, A(t)\) is the amount to be memorized in time \(t, M\) is the total amount to be memorized, and \(M-A\) is the amount remaining to be memorized. See Problems 25 and 26 in Exercises \(1.3\). (a) Since the DE is autonomous, use the phase portrait concept of Section \(2.1\) to find the limiting value of \(A(t)\) as \(t \rightarrow \infty\). Interpret the result. (b) Solve for \(A(t)\) subject to \(A(0)=0\). Sketch the graph of \(A(t)\) and verify your prediction in part (a).

Step-by-Step Solution

Answer

As \( t \to \infty \), \( A(t) \to \frac{k_1 M}{k_1 + k_2} \). The graph confirms this steady state value.

Step 1: Understanding the Differential Equation

The equation \( \frac{dA}{dt} = k_1(M - A) - k_2A \) models the rate of change of the amount memorized \( A \). \( k_1(M - A) \) represents acquisition, where \( M - A \) is the remaining amount to be memorized. \( k_2 A \) represents the forgetting aspect, where \( A \) is the amount already memorized.

Step 2: Setting Up the Autonomous Equation

The given DE is autonomous, meaning it doesn't explicitly depend on \( t \). The equation can be rearranged as \( \frac{dA}{dt} = k_1M - (k_1 + k_2)A \). This form shows that \( dA/dt \) is a linear function of \( A \).

Step 3: Analyzing Limiting Behavior

In steady state (as \( t \to \infty \)), the rate of change \( \frac{dA}{dt} = 0 \). Then, \( k_1M - (k_1 + k_2)A = 0 \), giving the limiting value \( A = \frac{k_1M}{k_1 + k_2} \). This value is where acquisition equals forgetting.

Step 4: Solving the Differential Equation for A(t)

To solve, recognize the DE as a first-order linear equation. The general solution is obtained by solving the homogeneous equation and adding a particular solution. Rewrite as \( \frac{dA}{dt} + (k_1 + k_2)A = k_1M \). An integrating factor \( \mu(t) = e^{(k_1 + k_2)t} \) is used. Integrating both sides results in \( A(t) = \frac{k_1M}{k_1 + k_2} + C'e^{-(k_1 + k_2)t} \).

Step 5: Applying Initial Condition

Given \( A(0) = 0 \), substitute in the solution: \[ 0 = \frac{k_1M}{k_1 + k_2} + C'e^{0} \] so \( C' = -\frac{k_1M}{k_1 + k_2} \). Substitute \( C' \) back: \( A(t) = \frac{k_1M}{k_1 + k_2} (1 - e^{-(k_1 + k_2)t}) \).

Step 6: Sketching the Graph and Interpretation

The graph of \( A(t) \) shows an S-shaped curve starting at 0 and asymptotically approaching \( \frac{k_1M}{k_1 + k_2} \). It shows that as time progresses, memorization balances between acquisition and forgetting, reaching a steady limit.

Key Concepts

Memorization Model

Understanding how we memorize information involves considering both acquiring and forgetting. In the given differential equation \( \frac{dA}{dt} = k_1(M - A) - k_2A \), we observe two main factors:

• Acquisition: Represented by \( k_1(M - A) \), it indicates how quickly we learn the remaining material \( M - A \).
• Forgetting: Modeled by \( k_2A \), it describes how we gradually forget what we've learned.

The model combines these two processes to predict the total memorization over time, balancing learning new information with the decay of already memorized content.

Phase Portrait

A phase portrait is a graphical illustration used to analyze the behavior of autonomous differential equations like ours. Since \( \frac{dA}{dt} \) does not depend on time directly, a phase portrait can depict how \( A(t) \) changes over time by analyzing the function's behavior solely concerning \( A \).

• On the vertical axis, plot \( A(t) \) which is the amount memorized.
• The horizontal axis represents the rate of change \( \frac{dA}{dt} \).

By studying this graph, we can identify equilibrium points where the rate of change is zero, and understand the general progression of memorization across time.

Autonomous Differential Equation

An autonomous differential equation does not explicitly include the independent variable, usually time \( t \), within its formula. Our equation \( \frac{dA}{dt} = k_1M - (k_1 + k_2)A \) is autonomous as \( t \) doesn’t appear directly. This property simplifies analysis because the behavior of the system depends entirely on the dependent variable, \( A\). Autonomous equations often enable us to find consistent behavior by evaluating steady states or equilibrium points without regard to specific times. This lets us focus on how the solution evolves inherently from the equation's structure.

Limiting Behavior

The limiting behavior of a differential equation tells us what happens to the system as time approaches infinity. For our equation, when \(t \to \infty \), \(\frac{dA}{dt} = 0\), signaling that the system reaches a steady state.
Solving for this gives \(A = \frac{k_1M}{k_1 + k_2}\). This is the equilibrium where learning and forgetting balance out, indicating the maximum amount we can memorize, factoring in forgetfulness.

• This balance point is crucial because it shows the ultimate capability of our memorization given constraints such as rate of forgetfulness.

Integrating Factor

The integrating factor is a mathematical technique used to solve first-order linear differential equations. It simplifies the process by transforming a non-exact differential equation into an exact one, making it easier to integrate.To solve \(\frac{dA}{dt} + (k_1 + k_2)A = k_1M\), we multiply through by the integrating factor \(\mu(t) = e^{(k_1 + k_2)t}\).
This technique allows us to integrate both sides easily, leading us to the solution \(A(t) = \frac{k_1M}{k_1 + k_2} + C'e^{-(k_1 + k_2)t}\), where \(C'\) is determined by initial conditions. This approach is powerful for finding specific solutions that describe real-world scenarios, such as the memorization model.