7.5 Approximating Solutions Using 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.

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.