Problem 42 - Advanced Engineering Mathematics

Question

Memorization 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

\(A(t)\) approaches \(\frac{k_1 M}{k_1 + k_2}\) as \(t \to \infty\). The function grows sigmoidally.

Step 1: Identify the Phase Portrait

To find the limiting value of \(A(t)\) as \(t \to \infty\), we use the given differential equation for memorization \( \frac{dA}{dt} = k_1(M - A) - k_2 A \). This is an autonomous differential equation. The equilibrium point occurs when \( \frac{dA}{dt} = 0 \). Solving, we have \( k_1(M - A) - k_2 A = 0 \). Rearranging gives \( A = \frac{k_1 M}{k_1 + k_2} \). As \(t \to \infty\), \(A(t)\) approaches this equilibrium value, \( \frac{k_1 M}{k_1 + k_2} \).

Step 2: Solve the Differential Equation

We solve the differential equation \( \frac{dA}{dt} = k_1(M - A) - k_2 A \) with the initial condition \( A(0) = 0 \). Separating variables, solve for \(A(t)\): \[ \frac{dA}{dt} + (k_1 + k_2)A = k_1 M \]. This is a first-order linear differential equation. The integrating factor is \( e^{(k_1 + k_2)t} \). Multiply through by the integrating factor and integrate both sides to obtain: \[ A(t) = \frac{k_1 M}{k_1 + k_2} (1 - e^{-(k_1 + k_2)t}) \].

Step 3: Interpret and Sketch Solution

The solution \(A(t) = \frac{k_1 M}{k_1 + k_2} (1 - e^{-(k_1 + k_2)t})\) shows that \(A(t)\) is an exponential function that grows from 0 at \(t = 0\) to \(\frac{k_1 M}{k_1 + k_2}\) as \(t \to \infty\). This means that without forgetting, the amount memorized rises quickly at first, then slows down as it approaches the equilibrium memorization value. The sketch of \(A(t)\) is an increasing curve asymptotically approaching \(\frac{k_1 M}{k_1 + k_2}\). This confirms the phase portrait analysis.

Key Concepts

Phase Portraits in Autonomous Differential Equations

In autonomous differential equations, the phase portrait is a useful tool to visually understand how a system behaves over time. For example, in the process of memorization, the differential equation \( \frac{dA}{dt} = k_1(M - A) - k_2 A \) describes how much material \(A\) is memorized at time \(t\). Autonomous differential equations depend only on the variable itself and time does not explicitly appear in them.

The phase portrait helps to identify equilibrium points and analyze stability. An equilibrium point is found when \( \frac{dA}{dt} = 0 \), which means the system doesn't change at that moment. Solving \( k_1(M - A) - k_2 A = 0 \) results in an equilibrium point at \( A = \frac{k_1 M}{k_1 + k_2} \).

As time \(t\) approaches infinity, the system's behavior tends towards these equilibrium values. In this memorization model, the amount memorized \(A(t)\) will eventually stabilize at this equilibrium. Visualizing the phase portrait offers insights into how quickly the memorization settles at this value and helps in predicting the system dynamics.

Understanding the Memorization Model

The memorization model considers the reality of forgetting material over time and is mathematically captured in this equation: \( \frac{dA}{dt} = k_{1}(M-A) - k_{2}A \).

• Here, \(A(t)\) is the amount memorized at time \(t\).
• \(M\) is the total amount to be memorized.
• \(k_1\) and \(k_2\) are positive constants representing the rate of memorization and forgetting, respectively.

The differential equation shows two competing actions: memorization (\(k_1(M-A)\)) and forgetting (\(-k_2A\)). The balance between these forces determines the amount memorized over time.

When a student begins studying, the focus is on the material yet to be memorized \((M - A)\). Over time, the forgetting term \(-k_2A\) becomes more significant as more of the material is initially memorized. This models the familiar experience where learning progresses rapidly at first and then slows. This mathematical description provides a deeper understanding of how knowledge acquisition can be affected by forgetting over time.

Solving First-Order Linear Differential Equations

Solving first-order linear differential equations involves a few clear steps that lead to understanding system behaviors in practical models like memorization. The general form is \( \frac{dA}{dt} + P(t)A = Q(t) \), where \(A\) depends on time \(t\).

In the memorization problem, it was \( \frac{dA}{dt} + (k_1 + k_2)A = k_1 M \). Solving such equations involves using an integrating factor, commonly expressed as \( e^{\int P(t)dt} \). This technique helps to simplify the equation, allowing integration and a solution for \(A(t)\).

For our memorization model, the integrating factor is \( e^{(k_1 + k_2)t} \). This allows the system to be rewritten and solved to find \( A(t) = \frac{k_1 M}{k_1 + k_2} (1 - e^{-(k_1 + k_2)t}) \).

Understanding this solution helps interpret how learning progresses over time, considering both the initial burst of learning and the gradual stabilization, confirming what we understand intuitively about the learning processes and the eventual memorization saturation.