Euler's Method

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):

$f\left(x\right)\approx f\left(a\right)+f\text{'}\left(a\right)(x-a)$

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

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 $y\text{'}=f(x,y)$. 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:

$y_{i+1}\approx y_i+hf(x_i,y_i)$

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 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$y_1\approx y_0+hf(x_0,y_0)$ 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.

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

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