7.5 Approximating Solutions Using Euler’s Method

What is Euler's Method?

Euler’s method is a way to find the numerical values of functions based on a given differential equation and an initial condition. We can approximate a function as a set of line segments using Euler’s method.

Before introducing this idea, it is necessary to understand two basic ideas.

This information allows us to do an algorithmic process to approximate function values when given a differential equation and an initial condition.

To showcase this method, let’s consider the following differential equation with a consequent initial condition:

Let’s say we want to approximate $y(7)$. We will create a table that essentially creates that line-segment link in Fig. 7.1:

Notice that we can fill in the rest of the table and continue the process to get closer and closer to $x = 7$. Also notice that the change in $x$ is a constant value (which is typically called the step-size).

We use the differential equation to find the slope at the given point and use Eq. 41 to find the change in $y$:

We can then find the new value of $y$ by adding the change in $y$ from the original $y$ value:

We can then fill in the rest of the table:

Note that as the step size approaches zero, the approximation becomes more and more exact. As an exercise, find an approximate value for $y(9)$. This means that $x = 7$ corresponds to $y = 249$, which is our approximate solution. 😃

Shorthand Summary of the Method

Euler's method is a first-order numerical procedure for approximating a solution to a differential equation. It is a simple and easy-to-implement method that is widely used in physics, engineering, and other fields.

Euler's method is based on the idea of approximating the solution curve of a differential equation by a sequence of straight lines. The method starts with an initial point on the solution curve, and then generates a sequence of points by moving along the tangent line at each point. The tangent line is determined by the slope of the solution curve at that point, which is given by the derivative of the solution function.

The basic procedure for Euler's method is as follows:

1. Start with an initial point $(x0, y0)$ on the solution curve.
2. Use the derivative of the solution function ($dy/dx = f(x, y)$) to find the slope at the initial point.
3. Use the slope to estimate the next point on the solution curve by moving a small step in the x-direction, called $h$, and adding the slope times $h$ to the y-value at the initial point.
4. Repeat steps 2 and 3 to generate a sequence of points on the solution curve. The main advantage of Euler's method is that it is simple to implement and understand. It is a good method for approximating a solution when an exact solution is not available. However, it is not very accurate and it may produce large errors when the step size is large.

Practice

Using Euler’s method, approximate the value of $y(2)$ using a step size of 0.25 given the following:

Then find the absolute error in the approximation by directly solving for $y(2)$  by using a calculator. Approximate $y(2)$ again using a step size of 0.2 and compare the absolute error in this approximation to the original absolute error.

Answer