Question

Euler's Method In Exercises \(73-78\) , use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use \(n\) steps of size \(h .\) $$ y^{\prime}=e^{x y}, \quad y(0)=1, \quad n=10, \quad h=0.1 $$

Short answer

Our final solution will be a table with \(n(10)\) values, each containing the x and y value computed at every step. Note, however, that this is an approximation of the solution and may not match the exact solution of the differential equation.

Step-by-step solution

Analyze Given Information

We are given the differential equation \(y'=e^{xy}\) with the initial condition y(0) = 1. We are to use Euler's method with 10 steps of size 0.1 to approximate the solution of the differential equation.

Initialize Parameters

Let's initialize our parameters. We set \(x_0 = 0\), \(y_0 = 1\), \(h=0.1\) and \(n=10\). \(x_0\) and \(y_0\) are our initial condition, \(h\) is our step size and \(n\) is the number of steps.

Apply Euler's Method

For each step from 1 to \(n=10\), we compute the new values of \(x_{i+1}\) and \(y_{i+1}\) using the formulas: \(x_{i+1} = x_i + h\) and \(y_{i+1} = y_i + h \cdot f(x_i, y_i)\), where \(f(x_i, y_i) = e^{x_i y_i}\). We repeat this process \(n\) times to create a table of approximate solutions for the differential equation.

Key concepts

Differential Equation

In mathematics, a differential equation is an equation that relates a function with one or more of its derivatives. In practical terms, these equations allow us to model the way quantities change over time, which is fundamental in fields such as physics, engineering, and economics.

For example, the equation \( y' = e^{xy} \) presented in the exercise is a first-order differential equation where \( y' \) denotes the derivative of \( y \) with respect to \( x \). The solution to a differential equation is a function, or a set of functions, that satisfies the equation for all values of \( x \) in a certain interval.

When dealing with such equations, it is often impossible to find an exact solution, and that's where numerical methods like Euler's Method come to play, allowing us to approximate the behavior of the function defined by the differential equation.

Initial Value Problem

An initial value problem in the context of differential equations is when you have a differential equation along with a specific value, or set of values, provided at the start. This 'initial value' allows us to find a unique solution to the differential equation that passes through the given point.

The problem in the exercise specifies the initial condition as \( y(0) = 1 \). This means that when \( x = 0 \) the value of \( y \) is 1. From this point, we begin to plot the trajectory of the function \( y \) as \( x \) increases, based on the rule provided by the differential equation. The initial value is crucial because, without it, there could be infinitely many functions that satisfy the differential equation without any way to determine which is the correct one to describe the situation we're modeling.

Numerical Approximation

Numerical approximation is the process of finding an approximate solution to problems that cannot be solved analytically or would be very difficult to solve exactly. This is especially important in differential equations, where exact solutions may not always be possible to obtain.

In the exercise, we use Euler's Method, which is a basic yet powerful tool for finding a numerical approximation to the solution of a differential equation. It works by moving step-by-step from the initial value, using the slope provided by the differential equation to estimate the rise over a small interval, known as the step size \( h \).

To illustrate, let's consider the specific case given in the exercise with \( h = 0.1 \) and 10 total steps (\( n = 10 \) steps). We begin at the initial value and calculate the next point by using the current point's slope and the step size. The idea is similar to following a path by taking regular strides in the direction indicated by the path. Over small intervals, this method provides a good approximation of the actual path, or in our case, the actual solution to the differential equation.

While Euler's Method is simple and intuitive, it is not without its drawbacks, such as accumulating error with each step. However, it is a starting point in numerical analysis and helps us understand more complex methods for approximating solutions to differential equations.