Question

Use Euler's method with step sizes \(h=0.05, h=0.025,\) and \(h=0.0125\) to find approximate values of the solution of the initial value problem $$ y^{\prime}=\frac{y^{2}+x y-x^{2}}{x^{2}}, \quad y(1)=2 $$ at \(x=1.0,1.05,1.10,1.15, \ldots, 1.5\). Compare these approximate values with the values of the exact solution $$ y=\frac{x\left(1+x^{2} / 3\right)}{1-x^{2} / 3} $$ obtained in Example 2.4.3. Present your results in a table like Table 3.1 .1 .

Short answer

Answer: The formula for Euler's method is given by \(y_{n+1} = y_n + h \cdot f(x_n, y_n)\), where \(y_{n+1}\) is the approximate value at \(x_{n+1}\), \(y_n\) is the current value at \(x_n\), \(h\) is the step size, and \(f(x_n, y_n)\) represents the right-hand side of the initial value problem.

Step-by-step solution

Define the function and the exact solution

Define the function \(f(x, y)\) as well as the exact solution of the initial value problem, \(y = g(x)\): \(f(x, y) = \frac{y^2 + xy - x^2}{x^2}\) \(g(x) = \frac{x\left(1+x^{2} / 3\right)}{1-x^{2} / 3}\)

Implement Euler's method for different step sizes

To apply Euler's method for the given function and step sizes, we need to repeat the following process for each step size \(h\): 1. Initialize the starting point \((x_0, y_0) = (1, 2)\) 2. Compute the new value of \(y_{n+1}\) according to Euler's method formula 3. Update the current position \((x_n, y_n)\) 4. Repeat step 2 and 3 until the desired \(x\) value is reached 5. Store the approximate value in a list for later comparison

Compute the exact solution at the same x-points

For each x-point reached while using Euler's method, compute the exact solution \(g(x)\) and store it together with the approximate values for the later comparison.

Present the results in a table format

Finally, organize the results into a table, including columns for the x-points, the exact solution, and each approximate solution for the different step sizes. This can be achieved with the help of a spreadsheet program or a programming language that supports tabular data manipulation.

Key concepts

Numerical Approximation

Numerical approximation is a powerful tool used to find approximate solutions to mathematical problems when an exact solution is difficult or impossible to obtain. In the context of differential equations, numerical approximation algorithms, like Euler’s method, allow us to estimate the value of a function at certain points using a simple iterative process.

Euler's method is particularly useful when dealing with initial value problems, where we have a starting point and wish to find out how variables change over time or across certain intervals. By approximating the function with small steps, the algorithm presents estimated solutions that become more accurate as the step size decreases.

This technique provides a balance between computational simplicity and accuracy. It is especially handy in scenarios where limited computational power is available, letting us assess the behavior and value of functions without requiring complex calculations.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation that comes with a given set of conditions at the start point of the interval. The aim is to determine the function that satisfies both the differential equation and the initial conditions.

For example, in the problem at hand, we have the differential equation:\[ y' = \frac{y^2 + xy - x^2}{x^2} \]
with the initial condition \(y(1) = 2\). This means we want to find a function \(y(x)\) that satisfies this equation and equals 2 when \(x=1\).

To tackle IVPs using numerical methods like Euler’s method, the differential equation gives us a way to calculate the slope at any given point. We then use this slope to estimate the value of the function at a small step away from the starting point, continuing this process iteratively to approximate the whole function over an interval.

Step Size

The step size, denoted as \(h\), in Euler's method refers to the interval between successive points where you are estimating the value of the solution. This is a critical component of the method, significantly influencing the accuracy of the approximation.

A smaller step size leads to more calculations, as it means you take smaller increments between each calculation, resulting in a more accurate approximation of the function. However, this comes at the cost of increased computational workload. Conversely, a larger step size reduces computational demand but may lead to less accurate results due to larger gaps between successive calculations.

In the exercise, using different step sizes (\(h = 0.05, 0.025, 0.0125\)) allows us to compare how precision and computational load balance against each other. A common practice in numerical analysis is to experiment with varying step sizes to ensure that the solution's accuracy is satisfying compared to the exact solution available, like in the given example.