Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=x^{2}, y(0)=0,[0,1] $$

Step-by-Step Solution

Answer

Using Euler's Method, the approximation of the solution at \( x = 1 \) is \( y \approx 0.24 \).

Step 1: Understand Euler's Method

Euler's Method is a numerical procedure to approximate solutions of first-order differential equations. Given the differential equation \( y' = f(x, y) \) and initial condition \( y(x_0) = y_0 \), with a step size \( h \), Euler's Method estimates the next value as \( y_{n+1} = y_n + h f(x_n, y_n) \).

Step 2: Identify Parameters

We are given the differential equation \( y'(x) = x^2 \) and the initial condition \( y(0) = 0 \). The step size \( h \) is 0.2, and we need to approximate the solution from \( x = 0 \) to \( x = 1 \).

Step 3: Compute the First Iteration

Starting at \( x_0 = 0 \), \( y_0 = 0 \), calculate \( y_1 \):\[y_1 = y_0 + h f(x_0, y_0) = 0 + 0.2(0^2) = 0\]Move to the next point \( x_1 = 0.2 \).

Step 4: Compute the Second Iteration

Using \( x_1 = 0.2 \) and \( y_1 = 0 \), calculate \( y_2 \):\[y_2 = y_1 + h f(x_1, y_1) = 0 + 0.2(0.2^2) = 0.008\]Move to \( x_2 = 0.4 \).

Step 5: Compute the Third Iteration

Using \( x_2 = 0.4 \) and \( y_2 = 0.008 \), calculate \( y_3 \):\[y_3 = y_2 + h f(x_2, y_2) = 0.008 + 0.2(0.4^2) = 0.04\]Move to \( x_3 = 0.6 \).

Step 6: Compute the Fourth Iteration

Using \( x_3 = 0.6 \) and \( y_3 = 0.04 \), calculate \( y_4 \):\[y_4 = y_3 + h f(x_3, y_3) = 0.04 + 0.2(0.6^2) = 0.112\]Move to \( x_4 = 0.8 \).

Step 7: Compute the Fifth Iteration

Using \( x_4 = 0.8 \) and \( y_4 = 0.112 \), calculate \( y_5 \):\[y_5 = y_4 + h f(x_4, y_4) = 0.112 + 0.2(0.8^2) = 0.24\]Move to \( x_5 = 1.0 \).

Step 8: Finalize Results

Finally, with \( x_5 = 1.0 \), we have \( y_5 = 0.24 \) as the approximate value of the solution using Euler's Method over the interval \([0,1]\).

Key Concepts

Euler's Method

Euler's Method is a fundamental technique employed in numerical analysis and is used to approximate solutions to differential equations. Imagine you're trying to predict the future position of a rocket based on its current velocity and position. Euler's Method works by taking small steps, called 'increments', in the direction of the differential equation's slope to estimate these future values.

The concept behind Euler's method is simple yet powerful. Given an initial point, you can determine approximation successively by following the formula:

• Start with an initial condition: the values of\( x_0 \) and\( y_0 \).
• Determine the slope or rate of change at this point using the function \( y' = f(x, y) \).
• Use a small step size \( h \) to predict the future value, by calculating: \( y_{n+1} = y_n + h \times f(x_n, y_n) \).

Euler's method is particularly useful when the actual solutions are difficult or impossible to find analytically, providing a numerical approximation instead.

Differential Equations

Differential equations are mathematical equations that describe how quantities change over time. They play a critical role across various fields such as physics, engineering, and economics.

In essence, a differential equation acts like a bridge that connects an initial condition with a function that changes over time.

• A first-order differential equation involves the first derivative of a function.
• For example, the differential equation presented in our problem is \( y' = x^2 \).

Differential equations can model natural phenomena, from population growth to the motion of planets, making them incredibly useful in scientific computations.

Initial Value Problem

An Initial Value Problem (IVP) is a specific type of differential equation that provides a starting point, called the "initial condition." This initial point tells us the value of the unknown function at a particular point.

Solving an IVP involves:

• Identifying a differential equation such as \( y' = f(x, y) \).
• Specifying the initial condition \( y(x_0) = y_0 \).
• Using methods like Euler's to find approximate solutions over a defined interval.

The IVP in this context is \( y(0) = 0 \), indicating that at \( x = 0 \), the value of \( y \) is 0. This sets the course for Euler’s Method to predict future values of \( y \) based on this starting point.

Numerical Approximation

Numerical approximation involves estimating the values of functions when exact solutions are not feasible. This method becomes crucial when a function is complex, or when no closed-form solution exists.

The role of numerical approximation is to provide practical solutions. Euler's Method serves as one such approach, wherein:

• The continuous problem is discretized using known values.
• Small, incremental calculations predict future values.
• This sequence of numbers approximates the actual function values over an interval.

In this example, we used Euler's Method to approximate \( y_x \) from \( x = 0 \) to \( x = 1 \) using a step size of \( h=0.2 \). Despite its simplicity, Euler's Method delivers reliable approximations for initial insights into more complex behavior of differential equations.