Question

You may want to save the results of these exercises, sincewe'll revisit in the next two sections.Use Euler's 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 wher the initial condition is imposed and \(i=1,2,3\). The purpose of these exercises is to familiarize you with the computational procedure of Euler's method. $$ y^{\prime}=y+\sqrt{x^{2}+y^{2}}, \quad y(0)=1 ; \quad h=0.1 $$

Short answer

Using Euler's method, we found the approximate values for the solution of the given initial value problem: 1. At \(x_1 = 0.1\), the approximation is \(y(0.1) \approx 1.2\). 2. At \(x_2 = 0.2\), the approximation is \(y(0.2) \approx 1.423\). 3. At \(x_3 = 0.3\), the approximation is \(y(0.3) \approx 1.667\).

Step-by-step solution

Write down the equation for Euler's method.

Euler's method is given by the following recursive formula: $$ y_{n+1} = y_n + hf(x_n, y_n) $$ Here, \(f(x, y) = y + \sqrt{x^2 + y^2}\) as given by the ODE.

Applying the initial conditions and step size to Euler's method formula.

The initial condition is \(y(0) = 1\), which implies \(y_0 = 1\), and the step size, \(h = 0.1\). Substituting these values in the recursive formula, we can calculate the approximation for each point.

Calculate the approximation at x1 (x0 + h).

We first calculate the approximation \(y_1\) at \(x_1 = 0.1\). Using the recursive formula: $$ y_1 = y_0 + h[f(x_0, y_0)] = 1 + 0.1[f(0, 1)] $$ Now, substitute \(x = 0\) and \(y = 1\) into \(f(x, y)\): $$ f(0, 1) = 1 + \sqrt{0^2 + 1^2} = 2 $$ So, the approximation at \(x1\) is: $$ y_1 = 1 + 0.1(2) = 1.2 $$

Calculate the approximation at x2 (x0 + 2h).

Next, calculate the approximation \(y_2\) at \(x_2 = 0.2\): $$ y_2 = y_1 + h[f(x_1, y_1)] = 1.2+ 0.1[f(0.1, 1.2)] $$ Substitute \(x = 0.1\) and \(y = 1.2\) into \(f(x, y)\): $$ f(0.1, 1.2) = 1.2 + \sqrt{0.1^2+1.2^2} \approx 2.227 $$ So, the approximation at \(x_2\) is: $$ y_2= 1.2 + 0.1(2.227) \approx 1.423 $$

Calculate the approximation at x3 (x0 + 3h).

Finally, calculate the approximation \(y_3\) at \(x_3 = 0.3\): $$ y_3 = y_2 + h[f(x_2, y_2)] = 1.423+ 0.1[f(0.2, 1.423)] $$ Substitute \(x = 0.2\) and \(y = 1.423\) into \(f(x, y)\): $$ f(0.2, 1.423) = 1.423 + \sqrt{0.2^2+1.423^2} \approx 2.440 $$ So, the approximation at \(x_3\) is: $$ y_3= 1.423 + 0.1(2.440) \approx 1.667 $$ Now we have approximate values using Euler's method for the solution of the given IVP at required points: $$ y(0.1) \approx 1.2 \\ y(0.2) \approx 1.423 \\ y(0.3) \approx 1.667 \\ $$

Key concepts

Initial Value Problem

An Initial Value Problem (IVP) is at the heart of many applications in science and engineering. It consists of a differential equation along with a specified value, called the initial condition, at a particular point. In our exercise, the IVP is defined by the differential equation \(y' = y + \root{x^2 + y^2}\), with the initial condition \(y(0)=1\). This sets the stage for determining how a quantity changes over time from a known starting point.

The central challenge in solving an IVP is to find a function \(y(x)\) that satisfies both the differential equation at any point \(x\) and the initial condition. While analytic solutions are ideal, they are not always possible, and numerical methods like Euler's method become invaluable tools in approximating the behavior of the solution near the initial point.

Differential Equations

Differential equations are mathematical equations that describe the relationships involving the rates at which quantities change. These are critical in depicting the behavior of physical systems. The equation \(y' = y + \root{x^2 + y^2}\) provided in our exercise is a first-order ordinary differential equation (ODE) where the rate of change of \(y\) with respect to \(x\) is related to both \(y\) and \(x\) itself.

There are various methods to tackle differential equations, including separation of variables, integrating factors, and series solutions. However, when the equation cannot be solved analytically, or an explicit solution is difficult to obtain, numerical methods like Euler's method provide an alternative means to approximate the solution over an interval.

Numerical Approximation

Numerical approximation entails the process of finding a numerical value that is close to the exact mathematical value, which may be difficult or impossible to calculate directly. Euler's method, showcased in this problem, is a type of numerical approximation that is particularly useful for solving ODEs. This method proceeds by creating a piecewise-linear approximation of the solution, stepping forward a small amount at a time.

The solution is built iteratively by moving from one point to the next using the slope provided by the differential equation. The step size \(h\), which was given as \(0.1\) in our exercise, determines the accuracy of the approximation – smaller values of \(h\) generally lead to more accurate approximations but require more computational steps.

Recursive Formula

A recursive formula represents a sequence of values where the next value is a function of one or more of the preceding values. In the context of the Euler method, the recursive formula is given by: \(y_{n+1} = y_n + hf(x_n, y_n)\). Here, \(y_{n+1}\) is the next value we want to find, \(y_n\) is the current value, and \(hf(x_n, y_n)\) is the product of the step size and the rate of change at the current point.

In our example, we use the recursive formula of Euler's method to calculate successive approximations \(y_1, y_2, y_3,\) and so on. This process allows us to 'step' through the problem domain, building up an approximate solution to the differential equation incrementally. By paying careful attention to the calculation at each step, students can improve their understanding of both the specific problem and the Euler method itself.