Problem 7 - Advanced Engineering Mathematics

Question

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use \(h=0.1\) and then use \(h=0.05\). \(-y\) $$ y^{\prime}=(x-y)^{2}, \quad y(0)=0.5 ; y(0.5) $$

Step-by-Step Solution

Answer

Using \( h = 0.1 \), \( y(0.5) \approx 0.5415 \); using \( h = 0.05 \), \( y(0.5) \approx 0.5413 \).

Step 1: Understand Euler's Method

Euler's method is a numerical approach to solving ordinary differential equations (ODEs). It involves using a given initial condition and a step size, denoted as \( h \), to estimate values of the function over an interval. The formula for Euler's method is: \[ y_{n+1} = y_n + h imes f(x_n, y_n) \]where \( f(x, y) = (x-y)^2 \) given in the problem.

Step 2: Setup Initial Conditions and Parameters

We have the initial condition \( y(0) = 0.5 \) and need to find \( y(0.5) \). We'll first calculate using a step size \( h = 0.1 \) and then \( h = 0.05 \). Start with \( x_0 = 0 \) and \( y_0 = 0.5 \).

Step 3: Calculate using \( h = 0.1 \)

For \( n = 0 \), we calculate:\[ x_0 = 0, \, y_0 = 0.5 \]\[ y_1 = y_0 + 0.1 imes (0 - 0.5)^2 = 0.5 + 0.01 = 0.51 \]For \( n = 1 \):\[ x_1 = 0.1, \, y_1 = 0.51 \]\[ y_2 = 0.51 + 0.1 imes (0.1 - 0.51)^2 = 0.51 + 0.01681 = 0.52681 \]Repeat this process until \( x_n = 0.5 \).

Step 4: Continue Calculation for \( h = 0.1 \)

Continuing from previous calculations:For \( n = 2 \):\[ x_2 = 0.2, \, y_2 = 0.52681 \]\[ y_3 = 0.52681 + 0.1 imes (0.2 - 0.52681)^2 = 0.53477239 \]For \( n = 3 \):\[ x_3 = 0.3, \, y_3 = 0.53477239 \]\[ y_4 = 0.53477239 + 0.1 imes (0.3 - 0.53477239)^2 = 0.53951863 \]For \( n = 4 \):\[ x_4 = 0.4, \, y_4 = 0.53951863 \]\[ y_5 = 0.53951863 + 0.1 imes (0.4 - 0.53951863)^2 = 0.54145625 \]

Step 5: Final Value for \( h = 0.1 \)

For \( n = 5 \), where \( x = 0.5 \):\[ x_5 = 0.5, \, y_5 = 0.54145625 \]The approximate value of \( y(0.5) \) using \( h = 0.1 \) is \( y(0.5) \approx 0.5415 \).

Step 6: Calculate using \( h = 0.05 \)

Now, let's use a finer step size \( h = 0.05 \). Start with initial values \( x_0 = 0, \, y_0 = 0.5 \) and calculate:For \( n = 1 \):\[ x_1 = 0.05, \, y_1 = 0.5 + 0.05 imes (0 - 0.5)^2 = 0.505 \]Continue this process until \( x_n = 0.5 \).

Step 7: Continue Calculation for \( h = 0.05 \)

Continue the calculations for each increment:For \( n = 2 \):\[ x_2 = 0.1, \, y_2 = 0.505 + 0.05 imes (0.1 - 0.505)^2 = 0.514025 \]For \( n = 3 \):\[ x_3 = 0.15, \, y_3 = 0.514025 + 0.05 imes (0.15 - 0.514025)^2 = 0.51474408 \]Repeat this process until \( x_n = 0.5 \).

Step 8: Final Value for \( h = 0.05 \)

Continuing from previous step:After following the calculations until \( x = 0.5 \), we find:\[ y(0.5) \approx 0.5413 \] using \( h = 0.05 \).

Step 9: Result Summary

With two different step sizes, we found:- Using \( h = 0.1 \), the approximate value of \( y(0.5) \) is 0.5415.- Using \( h = 0.05 \), the approximate value of \( y(0.5) \) is 0.5413.

Key Concepts

Euler's Method

Euler's Method is a simple yet powerful technique used in numerical methods to solve ordinary differential equations (ODEs). The method starts with a known initial value and uses a step-by-step approach to estimate the solution over an interval. It is particularly useful when an analytical solution is difficult or impossible to find. By incrementally advancing by a small step size, known as the step size \( h \), Euler's Method approximates the next value of the function using the formula:

• \( y_{n+1} = y_n + h \times f(x_n, y_n) \)

Here, \( f(x, y) \) represents the derivative of the function, expressing the rate at which \( y \) changes with respect to \( x \). By repeating this process iteratively, we construct an approximate solution to the differential equation over the desired interval.
Euler's Method is straightforward to understand and implement, making it an ideal introduction to numerical methods for solving differential equations. However, it can be less accurate compared to more advanced methods like the Runge-Kutta, particularly if the step size \( h \) is not chosen appropriately small.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations involving a function and its derivatives. These equations are "ordinary" because they involve derivatives with respect to only one independent variable, which typically represents time or space. In the context of mathematics and physics, ODEs are essential for modeling dynamic systems such as:

• Population growth over time
• Motion under gravitational force
• Electrical circuits

By understanding and solving ODEs, we can predict the behavior of such systems and make informed decisions based on the results. An ODE typically takes the form:

• \( y' = f(x, y) \)

This represents a mathematical relationship between the function \( y \), its derivative \( y' \), and the independent variable \( x \). Solving ODEs often involves finding the function \( y \) that satisfies this relationship given initial or boundary conditions.
In practice, solving ODEs analytically is sometimes challenging; hence numerical methods like Euler's Method come into play for approximations.

Initial Value Problems

Initial Value Problems (IVPs) are a class of ordinary differential equations (ODEs) together with specified values of the unknown function at a given point, called initial conditions. In an IVP, we are often interested in finding the solution of the ODE that exactly fits these conditions right from the start of the process.
The problem is structured as follows:

• \( y'(x) = f(x, y(x)) \)
• \( y(x_0) = y_0 \)

The initial condition \( y(x_0) = y_0 \) plays a crucial role as it serves as the starting point for methods like Euler's Method. The goal is to proceed from this known value using steps defined by the derivative equation to determine \( y(x) \) at subsequent values of \( x \).
Initial Value Problems are commonly used to model real-world phenomena where the starting state is known, such as the temperature of an object at \( t = 0 \) or the initial speed of a vehicle. By solving these problems, we gain insights into how the system evolves over time from that starting point.