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# 7.5 Approximating Solutions Using Eulerโs Method

Jacob Jeffries

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## What is Euler's Method?

Eulerโ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 Eulerโ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, letโs consider the following differential equation with a consequent initial condition:
Letโ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.

## Practice

Using Eulerโ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)ย  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.

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