Question

Using the Euler and the modified Euler methods, obtain approximate solutions of the initial-value problem $$ \begin{aligned} y^{\prime} &=x^{2}+y^{2} \\ y(0) &=1 \end{aligned} $$ at \(x=0.2\) with the step size \(h=0.1\).

Short answer

The approximate values of \(y(0.2)\) are: - Euler's method: \(y(0.2) \approx 1.221\) - Modified Euler's method: \(y(0.2) \approx 1.23725\)

Step-by-step solution

Set up Euler's Method formula

For Euler's method, we will use the following iterative formula: $$ y_{i+1} = y_{i} + hf(x_{i}, y_{i}), $$ where \(h\) is the step size, \(f(x,y)\) is the differential equation function, and \(y_{i+1}\) is the approximate value of \(y\) at the next step. In our case, \(f(x, y) = x^2 + y^2\).

Apply Euler's Method using step size h=0.1

Since we know that \(y(0) = 1\), we start with \(x_0 = 0\) and \(y_0 = 1\). We will apply Euler's method to find \(y(0.2)\) with step size \(h = 0.1\). We calculate: $$ y_{1} = y_{0} + h (x_0^2 + y_0^2) = 1 + 0.1 (0^2 + 1^2) = 1.1 $$ Next, we find \(y_2\) which is an approximation of \(y(0.2)\): $$ y_{2} = y_{1} + h (x_1^2 + y_1^2) = 1.1 + 0.1 (0.1^2 + 1.1^2) \approx 1.221 $$ Therefore, Euler's method gives an approximation of \(y(0.2) \approx 1.221\).

Set up Modified Euler's Method formula

Modified Euler's method (also known as Heun's method) uses the average of the function values at the start and end of the interval: $$ y_{i+1} = y_{i} + \frac{h}{2} [f(x_i, y_i) + f(x_{i+1}, y_{i} + hf(x_i, y_i))]. $$

Apply Modified Euler's Method using step size h=0.1

Now, let's use the Modified Euler method to find an approximation for \(y(0.2)\) with step size \(h = 0.1\). We have \(x_0 = 0\), \(y_0 = 1\), and \(f(x, y) = x^2 + y^2\). We calculate: $$ y_{1} = y_{0} + \frac{h}{2}(f(x_0, y_0) + f(x_1, y_0 + hf(x_0, y_0))) \approx 1.105 $$ Next, we find \(y_2\) which is an approximation of \(y(0.2)\): $$ y_{2} = y_{1} + \frac{h}{2}(f(x_1, y_1) + f(x_2, y_1 + hf(x_1, y_1))) \approx 1.23725 $$ Therefore, Modified Euler's method gives an approximation of \(y(0.2) \approx 1.23725\).

Compare the Results

We have obtained the following approximations for \(y(0.2)\): - Euler's method: \(y(0.2) \approx 1.221\) - Modified Euler's method: \(y(0.2) \approx 1.23725\) Both methods provide us with approximate solutions, but Modified Euler's method generally gives more accurate results as it takes into account the slope of the function at the beginning and the end of the interval.

Key concepts

Initial-Value Problem

An initial-value problem in the context of differential equations involves finding a function that satisfies a given ordinary differential equation (ODE) and adheres to specific starting conditions. Essentially, it sets the stage for the equation's trajectory by providing where to begin the journey.

For example, the initial-value problem presented in the exercise specifies the differential equation as \(y' = x^2 + y^2\) and the initial condition \(y(0) = 1\). Here, \(y'\) represents the derivative of \(y\) with respect to \(x\), and the challenge lies in determining the function \(y\) that complies with both the equation and the initial condition. Numerical methods, like Euler's and the modified Euler methods, are often employed when an analytical solution is difficult or impossible to find.

Modified Euler Method

The modified Euler method, also recognized as Heun's method, is a refinement of the basic Euler's approach for solving ODEs. It achieves greater accuracy by considering the average rate of change over an interval instead of just the rate of change at the interval's start.

To implement this method, as we have in the exercise, we use the formula \(y_{i+1} = y_{i} + \frac{h}{2} [f(x_i, y_i) + f(x_{i+1}, y_{i} + hf(x_i, y_i))]\), where \(h\) is the interval step size and \(f(x, y)\) represents the function derived from the differential equation. This method involves an initial prediction of \(y_{i+1}\) at the end of the interval, followed by an adjustment based on the average between the predicted and the initial function values. The result tends to be a closer approximation of the true solution than that provided by standard Euler's method.

Numerical Solutions of ODEs

Solving ordinary differential equations (ODEs) often necessitates numerical techniques, especially when the equations are complex or do not permit neat, analytical solutions. These numerical methods provide approximate solutions by discretizing the problem and iterating over small steps.

In the context of the given exercise, both Euler's and modified Euler's methods are numerical approaches that estimate the solution of the ODE at given points. Euler's method takes a forward step based on the slope (the derivative) at the current point, while the modified Euler method refines this by averaging the slopes at the current and the predicted next points. While neither method provides the exact solution, they both generate valuable estimates that can be made increasingly accurate by decreasing the step size \(h\).

Approximate Solutions

Approximate solutions are central to numerical methods for solving differential equations like the ones discussed. These solutions are practical when exact answers are elusive or when a quick estimation is necessary.

In practice, as shown through the exercise, the approximate solutions are calculated for discrete values of \(x\) using a chosen step size \(h\). The smaller the value of \(h\), the closer the approximation tends to be to the actual solution. However, this also means more computational steps. The trade-off between computational cost and accuracy is a key consideration when employing these methods. Moreover, initially small errors can accumulate with each step, potentially leading to significant deviations from the true solution over large intervals. Therefore, method selection and step size must be carefully balanced.

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They are vital in expressing a wide array of phenomena in science and engineering, encapsulating how change in one variable affects another.

The exercise deals with a simple form of a differential equation, where the change in the function \(y\) relies on both itself and another variable \(x\). To solve such an equation, one might seek an explicit formula, which is often difficult, or turn to numerical methods like Euler's and modified Euler's for approximate values. Such equations can be straightforward, like linear equations, or they can become highly involved, such as non-linear or partial differential equations. Each type may require a different approach or approximation method to find a solution that can predict real-world behavior with reasonable accuracy.