Problem 25 - Calculus Early Transcendentals

Question

Consider the initial-value problem \(y^{\prime}=y, y(0)=1\) and let \(y_{n}\) denote the approximation of \(y(1)\) using Euler's Method with \(n\) steps. (a) What would you conjecture is the exact value of \(\lim _{n \rightarrow+\infty} y_{n} ?\) Explain your reasoning. (b) Find an explicit formula for \(y_{n}\) and use it to verify your conjecture in part (a).

Step-by-Step Solution

Answer

The limit is \( e \) as \( n \to +\infty \). Euler's method approaches \( y(1) = e \).

Step 1: Understanding the Initial-Value Problem

The problem given is a differential equation with the condition \( y' = y \), and the initial value \( y(0) = 1 \). This is a typical exponential growth equation.

Step 2: Analyzing the Equation for the Exact Solution

The general solution to the differential equation \( y' = y \) is known to be \( y(t) = Ce^{t} \). Using the initial condition \( y(0) = 1 \), we find \( C = 1 \). Thus, the exact solution is \( y(t) = e^t \). When \( t = 1 \), this gives \( y(1) = e \).

Step 3: Setting Up Euler's Method

Euler's Method is a numerical method to approximate solutions to differential equations. Starting from \( y_0 = 1 \), we approximate \( y_n \) through steps. The update rule is \( y_{n+1} = y_n + h f(t_n, y_n) \), where \( f(t, y) = y \) and \( h = \frac{1}{n} \) because we want \( y_{n} \approx y(1) \).

Step 4: Derive the Explicit Formula for Euler's Method Approximation

Using \( f(t, y) = y \), the update becomes \( y_{n+1} = y_n + \frac{1}{n}y_n = y_n(1 + \frac{1}{n}) \). This iterates to the formula \( y_n = (1 + \frac{1}{n})^n \) since each step multiplies by \( 1 + \frac{1}{n} \).

Step 5: Evaluating the Limit

With \( y_n = (1 + \frac{1}{n})^n \), we recognize this as the limit definition of the exponential function \( e \). Therefore, as \( n \rightarrow +\infty \), \( y_n \rightarrow e \).

Step 6: State the Conclusion and Verify the Conjecture

The explicit formula \( y_n = (1 + \frac{1}{n})^n \) for Euler’s method approximation indeed approaches \( e \) as the number of steps increases. Thus, \( \lim_{n \rightarrow +\infty} y_n = e \). This verifies our conjecture from Step 2.

Key Concepts

Initial-Value Problem

An initial-value problem in mathematics involves finding a function that satisfies a differential equation and meets certain specified values at particular points, known as the initial conditions. For instance, in our exercise, the differential equation is given as \( y' = y \) with an initial condition \( y(0) = 1 \). This means we are tasked with finding the function \( y(t) \) that not only solves this differential equation but also equals 1 when \( t = 0 \).

Initial-value problems are crucial because they provide a unique solution to differential equations by anchoring the function to start at a known point. This serves as a critical starting point for applying numerical methods like Euler's Method. In practical terms, these problems often model real-world processes, such as how populations grow over time, by considering both the rules of change and the initial resources or conditions.

Differential Equations

Differential equations are equations that involve the derivatives of a function. In simple terms, they express how a quantity changes concerning another — primarily in terms of rates of change. Our given problem, \( y' = y \), tells us that the rate of change of \( y \) is equal to the value of \( y \) itself. This scenario implies a constant relative growth rate.

To solve a differential equation, we find a function that satisfies the given equation. In our exercise, the general solution is \( y(t) = Ce^{t} \), where \( C \) is a constant determined by the initial condition. These equations serve as the mathematical backbone for understanding and modeling dynamic systems in fields like physics, biology, and economics. By learning to manipulate and solve differential equations, students unlock a deeper understanding of how changing variables interact and evolve.

Exponential Growth

Exponential growth is a concept where the quantity grows at a rate proportional to its current value. In mathematical terms, if we have a differential equation \( y' = ky \), where \( k \) is a constant, the solution is an exponential function \( y(t) = Ce^{kt} \). This demonstrates how the quantity increases exponentially with time.

In our problem, \( y' = y \) suggests unit exponential growth, where the function grows by the factor of \( e \) over unit time intervals if no other constraints are applied. This idea of exponential growth is prevalent in various phenomena, including population growth, radioactive decay, and interest compounding, providing a realistic basis for predicting future values based on current conditions.

• Natural processes often follow exponential growth patterns when there are no limiting factors.
• Understanding exponential growth helps in planning and resource management as it predicts rapid increases over time.