Question

Use the improved Euler method to find approximate values of the solution of the given initial value problem at the points \(x_{i}=x_{0}+i h,\) where \(x_{0}\) is the point where the initial condition is imposed and \(i=1,2,3\). $$ y^{\prime}=y+\sqrt{x^{2}+y^{2}}, \quad y(0)=1 ; \quad h=0.1 $$

Short answer

Based on the Improved Euler method, the approximate values of the solution for the given initial value problem at different points are: 1. At \(x_1 = 0.1\), the approximate value is \(y_1 \approx 1.212013\) 2. At \(x_2 = 0.2\), the approximate value is \(y_2 \approx 1.459814\) 3. At \(x_3 = 0.3\), the approximate value is \(y_3 \approx 1.746468\)

Step-by-step solution

Understand the Improved Euler Method

Improved Euler method (also known as Heun's method) is a numerical method used to approximate the solution of differential equations. This method is an improvement over the standard Euler method as it incorporates an intermediate step that evaluates the average slope for more accurate results. The algorithm is: 1. Calculate the slope at the current point, called the "initial" slope. 2. Calculate a "predictor" value using this initial slope. 3. Calculate the slope at the predictor point, called the "corrector" slope. 4. Average the initial and corrector slopes. 5. Update the value of the unknown function using this average slope. Now let's apply the Improved Euler method to our specific problem.

Write down the given data

We have the following information given in the problem: - Differential equation: \(y^{\prime}=y+\sqrt{x^{2}+y^{2}}\) - Initial condition: \(y(0) = 1\) - Step size: \(h = 0.1\) - We need approximate values at \(x_{i} = x_0+ih\), where \(i = 1, 2, 3\) and \(x_0=0\).

Implement the Improved Euler Method

We'll now implement the Improved Euler Method algorithm for our specific problem: 1. Calculate the initial slope: \(k_{1} = y + \sqrt{x^{2}+y^{2}}\). Use the initial condition \(x_0 = 0\) and \(y_0 = 1\) 2. Calculate the predictor value: \(y_p = y_0 + h * k_{1}\) 3. Calculate corrector slope: \(k_{2} = y_p + \sqrt{(x_0+h)^{2}+(y_p)^{2}}\) 4. Average the initial and corrector slopes: \(k_{avg} = (k_{1} + k_{2})/2\) 5. Update the value of \(y\): \(y_{1} = y_0 + h * k_{avg}\) Repeat steps 1-5 for \(i = 2\) and \(i = 3\). Return the approximate values \(y_1\), \(y_2\), and \(y_3\). Now, let's calculate the approximate values step by step.

Calculate approximate values at \(x_i\)

At \(i=1\): 1. \(k_{1_1} = 1 + \sqrt{0+1^2} = 1 + 1 = 2\) 2. \(y_{p_1} = 1 + 0.1 * 2 = 1.2\) 3. \(k_{2_1} = 1.2 + \sqrt{0.1^2+1.2^2} \approx 2.240261\) 4. \(k_{avg_1} = (2+2.240261)/2 \approx 2.120131\) 5. \(y_1 = 1 + 0.1 * 2.120131 \approx 1.212013\) At \(i=2\): 1. \(k_{1_2} = 1.212013+\sqrt{0.1^2+1.212013^2} \approx 2.352288\) 2. \(y_{p_2} = 1.212013 + 0.1 * 2.352288 \approx 1.448642\) 3. \(k_{2_2} = 1.448642+\sqrt{0.2^2+1.448642^2} \approx 2.603735\) 4. \(k_{avg_2} = (2.352288+2.603735)/2 \approx 2.478011\) 5. \(y_2 = 1.212013 + 0.1 * 2.478011 \approx 1.459814\) At \(i=3\): 1. \(k_{1_3} = 1.459814+\sqrt{0.2^2+1.459814^2} \approx 2.724616\) 2. \(y_{p_3} = 1.459814 + 0.1 * 2.724616 \approx 1.732277\) 3. \(k_{2_3} = 1.732277+\sqrt{0.3^2+1.732277^2} \approx 3.028326\) 4. \(k_{avg_3} = (2.724616+3.028326)/2 \approx 2.876471\) 5. \(y_3 = 1.459814 + 0.1 * 2.876471 \approx 1.746468\)

Write down the results

We have successfully computed the approximate values of the solution for different points using the Improved Euler method: 1. \(y_1 \approx 1.212013\) 2. \(y_2 \approx 1.459814\) 3. \(y_3 \approx 1.746468\) These are our required approximate values for the given initial value problem.

Key concepts

Numerical Methods

Numerical methods are techniques used to find approximate solutions to mathematical problems that are difficult or impossible to solve analytically. These methods are particularly useful for solving problems in engineering, physics, and other sciences.
By using numerical methods, we can approximate the behavior of mathematical systems and make predictions. In this context, when we deal with complex differential equations that do not have straightforward solutions, numerical methods become essential tools.
There are several types of numerical methods, each with its approach and application area. Some common methods include the Euler method, Runge-Kutta methods, and the Improved Euler method, which provides more accuracy by considering average slopes.

Initial Value Problem

An initial value problem is a type of differential equation accompanied by a specific condition, known as the initial condition. This condition specifies the value of the unknown function at a particular point, allowing us to determine a unique solution.
The problem essentially involves finding a function that satisfies both the differential equation and the initial condition. In the given exercise, the differential equation is \(y' = y + \sqrt{x^2 + y^2}\) with the initial condition \(y(0) = 1\).
Initial value problems are very common in modeling real-world phenomena, where the initial conditions might represent the state of the system at the beginning of an observation or experiment.

Differential Equations

Differential equations are equations that involve a function and its derivatives. They are fundamental in describing various dynamic systems and processes in science and engineering.
These equations can be ordinary, involving derivatives with respect to a single variable, or partial, involving derivatives with respect to multiple variables. In this exercise, we are dealing with an ordinary differential equation (ODE).
Solving differential equations analytically might be challenging, but numerical methods provide practical alternatives to obtain solutions when conventional methods fail.

Heun's Method

Heun's Method, also known as the Improved Euler Method, is a sophisticated numerical technique used for solving ordinary differential equations. It enhances the basic Euler method by using two slope estimations, resulting in more accurate approximations.
The process is as follows: first, calculate the initial slope at the starting point of the interval. Next, predict the function value using this slope. Subsequently, a second slope is computed at this predicted value. Finally, the average of these two slopes provides a more precise estimate, improving accuracy compared to using a single slope.
In practical scenarios, Heun's Method is favored for its balance between simplicity and accuracy, making it a popular choice in the realm of numerical methods for initial value problems.