Euler's Method

Q: When can you use Euler's method?

Euler's Method can be used when the function f(x) does not grow too quickly.

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Given the complex nature of differential equations, these equations often cannot be solved exactly. However, there are numerous approximation algorithms for solving differential equations. One such algorithm is known as Euler's Method. Euler's Method relies on linear approximation as it uses a few small tangent lines derived based on a given initial value.

Katherine Johnson, one of the first African-American women to work as a scientist for NASA, used Euler's Method in 1961 to capacitate the first United States human space flight. Euler's Method allowed Johnson to estimate when the spacecraft should slow down to begin its descent into the atmosphere and resulted in a successful flight and landing!

The formula behind Euler's Method

Linear Approximation Review

The formula behind Euler's Method should be familiar to you. Recall the formula for linear approximation (can be found in the article Linear Approximations and Differentials) for f(x):

$f (x) \approx f (a) + f' (a) (x - a)$

where f(x) is the value of the function f at point x and a is a known initial value point.

Euler's Method linear approximation tangent line visualization StudySmarter The tangent line is formed from an initial point (a, f(a)) then the slope of the tangent line is used to approximate the value of f(y); here, the point (x, y) is the approximation while the point (x, f(y)) is the actual value - StudySmarter Original

Euler's Method Formula

Similarly, the general formula for Euler's Method for a differential equation of the form $y' = f (x, y)$ . The only difference between Euler's method and linear approximation is that Euler's method uses multiple approximation iterations to find a more exact value. Using Euler's method, we use x₀ and y₀, which are typically given as initial values, to estimate the slope of the tangent at x₁. It looks like this:

$y_{i + 1} \approx y_{i} + h f (x_{i}, y_{i})$

where $y_{i + 1}$ is the next solution value approximation, $y_{i}$ is the current value, $h$ is the interval between steps, and $f (x_{i}, y_{i})$ is the value of the differential equation evaluated at $(x_{i}, y_{i})$ .

Let's break this formula down further.

Derivation of Euler's Method

Consider the picture below.

Euler's Method approximation graph StudySmarter Euler's Method General Formula Intuition - StudySmarter Original

With an initial point $(x_{0}, y_{0})$ , we can find a tangent line with a slope of $f (x_{0}, y_{0})$ . We can use these values to approximate the point $(x_{1}, y_{1})$ where $x_{1} = x_{0} + h$ and $y_{1} \approx y_{0} + h f (x_{0}, y_{0})$ according to basic coordinate geometry. This operation can be done as many times as need be. However, it's important to mention that using a smaller step size h will produce a more accurate approximation. A larger step size h will produce a less accurate approximation.

If y₁ is a good approximation, then using Euler's method will give us a good estimate of the actual solution. However, if y₁ is not a good approximation, then the solution using this method will be off as well!

Importance of Euler's Method

Differential equations are commonly used to describe natural phenomena in the natural world with applications ranging in simplicity from the movement of a car to spacecraft trajectory models. Unfortunately, these equations cannot be solved directly given their complexity. This is where Euler's Method and other differential equation approximation algorithms come in. We can use differential equation approximation algorithms, like Euler's Method, to find an approximate solution. An approximate solution is much better than no solution at all!

Limitations of Euler's Method

Though Euler's Method is a simple and direct algorithm, it is less accurate than many algorithms like it. As previously mentioned, using a smaller step size h can increase accuracy but it requires more iterations and thus an unreasonably larger computational time. For this reason, Euler's Method is rarely used in practice. However, Euler's Method forms a basis for more accurate and useful approximation algorithms.

Euler's Method Examples

A step-by-step method

Consider the differential equation $\frac{d y}{d x} = 6 - 2 \frac{y}{x}$ with an initial value of $y (3) = 1$ . Use $h = 0.2$ to approximate $y (4)$ .

Step 1: Find the slope of the tangent line at the initial point

To find the tangential slope at $(3, 1)$ , we simply plug it into the differential equation to get

$\frac{d y}{d x} = 6 - 2 (\frac{1}{3}) = \frac{16}{3}$

Step 2: Find our new x-value

To find our next x-value, we add h to the initial x-value to get

$x_{1} = 3 + \frac{1}{5} = \frac{16}{5}$

Step 3: Plug in our values to obtain our new y-value approximation

So, we have:

Step size, $h = 0.2 = \frac{1}{5}$
Initial y-value, $y_{0} = 1$
The slope of the tangent line at the initial value, $f (x_{0}, y_{0}) = \frac{16}{3}$

Plugging in all of our values, we get

$\begin{array}{rcl} y_{1} & \approx & y_{0} + h \cdot f (x_{0}, y_{0}) \\ y_{1} & \approx & 1 + (\frac{1}{5}) (\frac{16}{3}) \\ \approx & 1 + \frac{16}{15} \\ \approx & \frac{31}{15} \end{array}$

So, the approximation to the solution at $x = 3 + 0.2 = 3.2$ is $\frac{31}{15}$ or

$y (3.2) \approx \frac{31}{15}$

Step 4: Repeat the algorithm as many times as needed to obtain y(4)

Given that our step size is 0.2, we will have to repeat the algorithm 4 more times:

Using $(\frac{16}{5}, \frac{31}{15})$ : $f (\frac{16}{5}, \frac{31}{15}) = 6 - 2 \frac{\frac{31}{15}}{\frac{16}{5}} = \frac{113}{24}, x_{2} = \frac{17}{5}, y_{2} \approx \frac{31}{15} + (\frac{1}{5}) (\frac{113}{24}) = \frac{361}{120}$
Using $(\frac{17}{5}, \frac{361}{120})$ : $f (\frac{17}{5}, \frac{361}{120}) = 6 - 2 \frac{\frac{361}{120}}{\frac{17}{5}} = \frac{863}{204}, x_{3} = \frac{18}{5}, y_{3} \approx \frac{361}{120} + (\frac{1}{5}) (\frac{863}{204}) = \frac{2621}{680}$
Using $f (\frac{18}{5}, \frac{2621}{680})$ : $f (\frac{18}{5}, \frac{2621}{680}) = 6 - 2 \frac{\frac{2621}{680}}{\frac{18}{5}} = \frac{4723}{1224}, x_{4} = \frac{19}{5}, y_{4} \approx \frac{2621}{680} + (\frac{1}{5}) (\frac{4723}{1224}) = \frac{3539}{765}$
Using $f (\frac{19}{5}, \frac{3539}{765})$ : $f (\frac{19}{5}, \frac{3539}{765}) = 6 - 2 \frac{\frac{3539}{765}}{\frac{19}{5}} = \frac{10364}{2907}, x_{5} = 4, y_{5} \approx \frac{3539}{765} + (\frac{1}{5}) (\frac{10364}{2907}) = \frac{913}{171}$

Finally, we have obtained our approximation at $y (4) \approx \frac{913}{171} \approx 5.339$ !

When solving multiple iterations of Euler's Method, it may be useful to construct a table for each of your values! In iterative problems such as these, tables can help to our numbers organized.

For this problem, a table might look like:

(x_i, y_i)	dy/dx	h = 0.2	x_i+1	y_i+1
$(3, 1)$	$\frac{16}{3}$		$\frac{16}{5}$	$\frac{31}{15}$
$(\frac{16}{5}, \frac{31}{15})$	$\frac{113}{24}$		$\frac{17}{5}$	$\frac{361}{120}$
$(\frac{17}{5}, \frac{361}{120})$	$\frac{863}{204}$		$\frac{18}{5}$	$\frac{2621}{680}$
$(\frac{18}{5}, \frac{2621}{680})$	$\frac{4723}{1224}$		$\frac{19}{5}$	$\frac{3539}{765}$
$(\frac{19}{5}, \frac{3539}{765})$	$\frac{10364}{2907}$		$4$	$\frac{913}{171}$

Step 5: Check the error

As this specific example can be solved directly, we can check the global error of our answer.

The direct solution to the differential equation is $y = \frac{- 45}{x^{2}} + 2 x$ . Plugging in x = 4, we get

$\begin{array}{rcl} y & = & \frac{- 45}{16} + 8 \\ = & \frac{83}{16} \\ = & 5.1875 \end{array}$

To check the percent error, we simply compute

$\begin{array}{rcl} % e r r o r & = & \frac{| e x a c t - a p p r o x i m a t i o n |}{e x a c t} \times 100 % \\ = & \frac{|\frac{83}{16} - \frac{913}{171}|}{\frac{83}{16}} \times 100 \\ \approx & 2.92 % \end{array}$

Our error is relatively low!

We use absolute values in the percent error calculation because we don't care if our approximation is above or below the actual value, we just want to know how far away it is!

Lucky for us, all Euler's Method problems follow the same simple algorithm.

Euler's Method - Key takeaways

Euler's Method is an approximation tool for differential equation solving based on linear approximation
The general Euler's Method formula is yi+1≈yi+h·f(xi, yi)where
- $y_{i + 1}$ is the next solution value approximation,
- $y_{i}$ is the current value,
- $h$ is the interval between steps, and
- $f (x_{i}, y_{i})$ is the value of the differential equation evaluated at $(x_{i}, y_{i})$
Euler's Method is rarely used in real-world applications as the algorithm tends to have low accuracy and requires vast computation time

Frequently Asked Questions about Euler's Method

Euler's Method can be used when the function f(x) does not grow too quickly.

Euler's Method is an approximation tool for differential equation solving based on linear approximation.

The Euler's Method formula is based on the formula for linear approximation. The next approximation is the sum of the old approximation value and the product of the step size and the differential equation at the old point.

Euler's Method is used for approximating solutions to differential equations that cannot be solved directly.

Euler's Method is important because most differential equations cannot be solved directly and thus must be estimated through approximation.

Flashcards in Euler's Method4 Start learning

What is Euler's Method based on?

Linear approximation using a tangent line

What are the limitations of Euler's Method?

Low accuracy
Large computational time

Explain Euler's Method in words.

Euler's Method goes through an arbitrary number of iterations where each iteration uses the slope of an approximated tangent line to estimate where the next point will be.

A larger step value h produces a ____ accurate approximation while a smaller step value h produces a ____ accurate approximation.

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