Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Apply Euler's Method to the equation \(y^{\prime}=y, y(0)=1\) with an arbitrary step size \(h=1 / N\) where \(N\) is a positive integer. (a) Derive the relationship \(y_{n}=(1+h)^{n}\). (b) Explain why \(y_{N}\) is an approximation to \(e\).

Step-by-Step Solution

Answer

(a) \( y_n = (1 + h)^n \). (b) \( y_N = (1 + \frac{1}{N})^N \) approximates \( e \) as \( N \to \infty \).

Step 1: Understand Euler's Method

Euler's Method is a numerical technique to approximate solutions of differential equations. For the differential equation \( y' = y \) with initial condition \( y(0) = 1 \), and a step size \( h \), we calculate \( y_{n+1} = y_n + h y'(x_n, y_n) = y_n + h y_n \).

Step 2: Apply Step Increment

Starting from the initial condition \( y_0 = 1 \), apply the increment calculation: \( y_1 = y_0 + h y_0 = 1 + h \), \( y_2 = y_1 + hy_1 = y_1(1 + h) = (1 + h)^2 \), and continue this pattern.

Step 3: Generalize to Derive Relationship

Notice the pattern that emerges: \( y_1 = (1 + h) \), \( y_2 = (1 + h)^2 \), and similarly, at each step \( n \), \( y_n = (1 + h)^n \). This supports the derived relationship \( y_n = (1 + h)^n \).

Step 4: Substitute for y_N

Use the expression \( y_n = (1 + h)^n \) when \( n = N \), resulting in \( y_N = (1 + \frac{1}{N})^N \). This gives the approximation \( y_N \) for the given problem.

Step 5: Approximate e by Limit

As \( N \to \infty \), the expression \( \left(1 + \frac{1}{N}\right)^N \) approaches the mathematical constant \( e \). Therefore, \( y_N = \left(1 + \frac{1}{N}\right)^N \) serves as an approximation for \( e \).

Key Concepts

Differential Equations

Differential equations form the backbone of many natural processes. They are equations that involve an unknown function and its derivatives. These equations express how a quantity changes over time or space. In simple terms, they provide the rules for how things evolve. For instance, in the exercise given, the differential equation is \( y' = y \). This tells us that the rate of change of \( y \) is equal to the value of \( y \) itself. Such equations appear in a variety of situations, such as modeling population growth, cooling of objects, and current flow in electrical circuits. This specific equation is a first-order differential equation and can be solved analytically or numerically. Analytically, it represents exponential growth. Numerically, methods like Euler's Method help us find approximate solutions.

Numerical Methods

Numerical methods are techniques used to approximate solutions to mathematical problems. These methods are essential when exact solutions are difficult or impossible to find. Euler's Method, a type of numerical method, is particularly used for solving differential equations numerically.Euler's Method works by taking small steps from an initial value, using the derivative to estimate the next value. In the exercise, with an initial condition \( y(0) = 1 \) and step size \( h \), Euler's formula is applied iteratively:

• Start with \( y_0 = 1 \).
• Calculate \( y_{n+1} = y_n + h y_n \) which simplifies to \( y_{n+1} = y_n(1 + h) \).
• This process constructs the sequence: \( y_1 = (1 + h), y_2 = (1 + h)^2, y_3 = (1 + h)^3 \), and so on.

The pattern reveals the relationship \( y_n = (1 + h)^n \), highlighting how Euler's Method provides a step-by-step approximation of the solution.

Approximation of e

Euler's number \( e \) is a mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and arises naturally in processes described by continuous growth or decay. In the exercise, Euler's Method is used with step size \( h = \frac{1}{N} \). When reaching the \( N \)-th step, we have \( y_N = \left(1 + \frac{1}{N}\right)^N \). This expression is significant because, as \( N \to \infty \), \( \left(1 + \frac{1}{N}\right)^N \) approaches the value of \( e \). Therefore, using larger \( N \) in the method allows us to approach \( e \) very closely. This is not just an interesting approximation, but also a fundamental concept in understanding exponential growth and continuous compounding in mathematics and its applications.