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7.5 Approximating Solutions Using Eulerโ€™s Method

#euler'smethod

#differentialequations

โฑ๏ธย ย 2 min read

written by

Jacob Jeffries

jacob jeffries

June 8, 2020


https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(565).png?alt=media&token=dea976c0-1d49-43a4-9c88-73d7245f7992

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.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(617).png?alt=media&token=fe5e7a85-6180-4b84-ac55-e8edee4b48d2

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

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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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(624).png?alt=media&token=6908c593-cd3a-4465-b35e-26544551d7cb

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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(625).png?alt=media&token=967695d0-5019-4e66-ac5b-ba178682ea10

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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(626).png?alt=media&token=35ec439b-0701-4cc9-9171-9392c815ac2f

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(627).png?alt=media&token=f276e25e-b950-41c1-9e12-f1336e290bfb

We can then fill in the rest of the table:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(628).png?alt=media&token=0dbd1426-c679-4401-a8d2-6fd2d13a1d4a

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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(630).png?alt=media&token=bb6423bf-1b06-46c1-8e6a-74fb4b3df780

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.

Answer

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(634).png?alt=media&token=58cc3bbc-2e7e-417f-a3fa-23eba0423d45

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(635).png?alt=media&token=24e5c269-3f57-452e-9b3e-7f4a4a1ae079

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(636).png?alt=media&token=21bb1004-1a1c-40e6-a7a0-75805927530b

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