Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-16, use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=-y, y(0)=2,[0,1] $$

Step-by-Step Solution

Answer

The approximate solution at \( x = 1.0 \) is \( y = 0.65536 \).

Step 1: Understanding Euler's Method

Euler's Method is a numerical technique to find approximate solutions to differential equations of the form \( y' = f(x, y) \). It iteratively updates the solution using the formula: \( y_{n+1} = y_n + h imes f(x_n, y_n) \), where \( h \) is the step size.

Step 2: Initialize Variables

Given the problem, start with the initial condition \( y(0) = 2 \). Set \( x_0 = 0 \) and \( y_0 = 2 \). The step size \( h \) is given as 0.2.

Step 3: Calculate for x = 0.2

Use Euler's Method to calculate the next value: \( y_1 = y_0 + h imes (-y_0) = 2 + 0.2 imes (-2) = 2 - 0.4 = 1.6 \). Thus, for \( x_1 = 0.2 \), \( y_1 = 1.6 \).

Step 4: Calculate for x = 0.4

Compute \( y_2 = y_1 + h imes (-y_1) = 1.6 + 0.2 imes (-1.6) = 1.6 - 0.32 = 1.28 \). For \( x_2 = 0.4 \), \( y_2 = 1.28 \).

Step 5: Calculate for x = 0.6

Compute \( y_3 = y_2 + h imes (-y_2) = 1.28 + 0.2 imes (-1.28) = 1.28 - 0.256 = 1.024 \). For \( x_3 = 0.6 \), \( y_3 = 1.024 \).

Step 6: Calculate for x = 0.8

Compute \( y_4 = y_3 + h imes (-y_3) = 1.024 + 0.2 imes (-1.024) = 1.024 - 0.2048 = 0.8192 \). For \( x_4 = 0.8 \), \( y_4 = 0.8192 \).

Step 7: Calculate for x = 1.0

Compute \( y_5 = y_4 + h imes (-y_4) = 0.8192 + 0.2 imes (-0.8192) = 0.8192 - 0.16384 = 0.65536 \). For \( x_5 = 1.0 \), \( y_5 = 0.65536 \).

Key Concepts

Numerical Approximation

Numerical approximation is a crucial tool in mathematics, especially when solving complex problems where an exact analytical solution is difficult to achieve or does not exist. It involves finding approximate solutions by using simpler arithmetic calculations.

One common method of numerical approximation is Euler's Method. This approach helps us approximate solutions to differential equations by making small steps along the function graph, rather than attempting to solve the equation exactly. This is particularly useful for differential equations that do not have simple, closed-form solutions.

• To use Euler's Method, you need an initial value, a specified interval, and a chosen step size, denoted as \( h \).
• Euler’s formula, \( y_{n+1} = y_n + h \times f(x_n, y_n) \), allows you to iteratively find the next point.

By progressively applying these small steps, you construct an approximate curve that represents the solution to the differential equation. Though it's approximate, it provides a practical way to understand complex systems.

Differential Equations

Differential equations are equations that involve an unknown function and its derivatives. They play a fundamental role in describing various phenomena in engineering, physics, economics, biology, and more.

For example, a simple differential equation can express the rate at which a population grows or decays. The equation \( y' = -y \) from our exercise tells us that the rate of change of the function \( y \) is proportional to the negative of the function itself. This is known as an exponential decay differential equation.

• Differential equations are categorized based on the highest order of derivative present and the linearity of the equation.
• They can be solved by analytical methods, yielding exact solutions, or by numerical methods, like Euler's Method, for approximate solutions.

Understanding differential equations is essential because they model real-world systems and help predict their behavior under various conditions.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation accompanied by specified initial conditions. It is crucial because it defines the starting point from which a solution is sought.

In practice, the initial condition \( y(x_0) = y_0 \) specifies the value of the solution at a particular point \( x_0 \). In our exercise, this is given as \( y(0) = 2 \), which means when \( x = 0 \), \( y \) starts at 2. This problem-solving technique is common for modeling time-evolving systems where the initial state or condition heavily influences future behavior.

• IVPs are solved by integrating the differential equation with respect to the initial conditions.
• Essentially, an initial condition provides the specific solution to the differential equation from the family of solutions that exist.

Understanding initial value problems allows us to capture the dynamics of systems accurately and predict future states based on initial observations.