Question

Euler's Method In Exercises \(79-81\) , complete the table using the exact solution of the differential equation and two approximations obtained using Euler's Method to approximate the particular solution of the differential equation. Use \(h=0.2\) and \(h=0.1,\) and compute each approximation to four decimal places. $$ \frac{d y}{d x}=\frac{2 x}{y} \quad(0,2) \quad y=\sqrt{2 x^{2}+4} $$

Short answer

By applying the Euler's method to the given differential equation with two different step sizes, tables of numerical approximations and exact solution can be calculated. More accurate results are expected to be seen with smaller step sizes, i.e., h=0.1, offering a concrete understanding of numerical methods and their relevance in solving differential equations.

Step-by-step solution

Understand the Euler's Method

The Euler's method is a numerical procedure used to approximate the solutions to initial value problems of the form dy/dx = f(x,y) with an initial condition y(x0)=y0. It progresses iteratively by the formula: y_n+1 = y_n + h * f(x_n, y_n). For the problem at hand, f(x,y) = 2*x/y.

Apply Euler's Method for h = 0.2

Plot the table with three columns: x, y_exact, and y_approx for h = 0.2. The initial conditions are x_0 = 0 and y_0 = 2. We use the iterative formula y_n+1 = y_n + h * f(x_n, y_n) to calculate the next y. f(x, y) is given as 2*x/y. Repeat the procedure until it reaches the final x value.

Apply Euler's Method for h = 0.1

Next, do the analogous procedure, but this time for the step size h = 0.1. As a smaller step size, it should provide a more accurate approximation to the exact solution.

Compute the Exact Solution

The exact solution is given by the function y = sqrt(2*x^2+4). Calculate this for every x value in the table, comparing the results with the approximations from the previous steps.

Key concepts

Differential Equations

Differential equations are mathematical tools that describe the relationship between a function and its derivatives. They express how a quantity changes in response to another. In the context of our exercise, we are dealing with a specific type of differential equation known as an ordinary differential equation (ODE), represented as \( \frac{dy}{dx} = \frac{2x}{y} \). This particular equation is a way to describe a rate of change where the derivative of the function \( y \) with respect to \( x \) depends on both \( x \) and \( y \) itself.

Solving differential equations is critical in various scientific fields, such as physics, engineering, and economics, because they model real-world phenomena. The exact solution of the given ODE is represented by \( y = \sqrt{2x^2 + 4} \), which we will use to compare against our approximate solutions obtained through Euler's Method. Understanding the exact solutions can provide deeper insights into the behavior of the system being analyzed.

Numerical Methods

Numerical methods are techniques used to obtain approximate solutions to mathematical problems that may not be easily solvable through analytical methods. Adopting a numerical approach is often necessary when dealing with complex differential equations or when an exact solution is too difficult or impossible to find.

In our exercise, Euler's Method is the numerical method chosen to approximate the solution of the given differential equation. It's one of the simplest methods to implement and provides an introduction to the idea of iterative solutions. Despite its simplicity, Euler's Method can offer valuable approximations and insights into the nature of the solution, especially when the step size \( h \) is sufficiently small. This approach is a stepping stone towards more sophisticated methods, like the Runge-Kutta Methods or the Adams-Bashforth Methods, which can provide more accurate results using similar iterative techniques.

Initial Value Problems

An initial value problem (IVP) is a type of differential equation along with a specified value, called the initial condition, at a given point. The goal is to find a function that not only satisfies the differential equation but also passes through the given initial point. For our exercise, the IVP is formulated with the differential equation \( \frac{dy}{dx} = \frac{2x}{y} \) and the initial condition provided at the point \( (0, 2) \).

IVPs are pervasive in fields such as physics, where initial conditions can be thought of as the state of the system at the beginning of an experiment or the moment of observation. Successfully determining the function that describes the system is central to predicting its future behavior. By utilizing Euler's Method, an approximation path is constructed step-by-step from the initial condition, approximating how the system evolves from that specific starting point.

Approximation of Solutions

Approximation of solutions is the process of finding a value that is close to the true solution of a problem but simpler to obtain. In numerical analysis, this typically involves using methods that incrementally build a solution through a series of steps, with each step improving the accuracy of the approximation.

In our exercise, Euler's Method approximates the solution of the ODE by starting at the initial condition and iteratively applying the formula \( y_{n+1} = y_n + h \times f(x_n, y_n) \). By reducing the step size \( h \) from 0.2 to 0.1, we effectively increase the precision of our approximation, trading off increased computation for improved accuracy. Students must compare these approximate values to the exact solution to understand the level of error and gain confidence in using numerical methods for different step sizes. Additionally, appreciating the difference in results provided by varying \( h \) can lead to a better grasp of how numerical methods behave as approximations.