How To Do Euler’s Method?-Simply Explained In A Powerful Example

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Yes, calculus is no doubt considered the heart of mathematics. If we take a look at the historical background of calculus, we will see many amendments that have been made to resolve various complex problems. An Euler’s method is one such technique that is used to simplify the ordinary differential equations. Also, you can make use of a free online Euler’s method calculator to perform instant computations in no time. You can resolve various differential equations with this differential equation calculator easily.

 

Anyways, in this article, we will be discussing the proper way to carry out calculations using Euler’s method.

 

Stay focused!

How To Do Euler’s Method?

This particular method is considered the most clear-cut way for the numerical integration of the ordinary differential equation. As there exist many various ways to differentiate a differential equation, but Euler’s method is considered the best and most reliable among all. 

Euler’s Method Formula:

You can integrate any differential equation with the help of the following Euler’s method formula which is also used by the free Euler’s method calculator.

 

$$A_n = A_{n-1} + hA \left(B_{n-1}, A_{n-1}\right)$$

Example Problem:

Consider the following differential equation below:

 

$$ y’ + 2y = 2 – e^{-4t} \hspace{0.25in} y\left(0\right) $$

 

Find approximate values of the solution for t = 0.1, 0.2, 0.3, 0.4, and 0.5 using Euler’s Method with a step size of h = 0.1. Compare them to the solution’s exact values at these locations.

 

Solution:

We’ll leave it to you to confirm that the answer is correct because this is a reasonably easy linear differential equation and is written a follows:

 

$$ y\left(t\right) = 1 + \frac{1}{2}e^{-4t} – \frac{1}{2}e^{-2t} $$

 

We must first rewrite the differential equation in the format below in order to employ Euler’s Method.

 

$$ y’ = 2 – e^{-4t} -2y $$

 At this point, we can assume that:

 

$$ f\left(t, y\right) = 2 – e^{-4t} -2y $$

And also \(t_{0} = 0\) and \(y_{0} = 1\). Keep in mind that putting all these values in the free online eulers method calculator will display precise and instant outputs on your screen.

 

Now, this is the time to carry out computations.

 

$$ \begin{align*}{f_0} &= f\left( {0,1} \right) = 2 – {{\bf{e}}^{ – 4\left( 0 \right)}} – 2\left( 1 \right) =  – 1\\ {y_1} & = {y_0} + h\,{f_0} = 1 + \left( {0.1} \right)\left( { – 1} \right) = 0.9\end{align*} $$

As a result, \(y_{1} = 0.9\) is an approximation to the solution at \(t_{1} = 0.1\).

 

After doing so, the next step includes:

 

$$ \begin{align*}{f_1} & = f\left( {0.1,0.9} \right) = 2 – {{\bf{e}}^{ – 4\left( {0.1} \right)}} – 2\left( {0.9} \right) =  – \,0.470320046\\  {y_2} & = {y_1} + h\,{f_1} = 0.9 + \left( {0.1} \right)\left( { – \,0.470320046} \right) = 0.852967995\end{align*} $$

 

As a result, at \(t_{2} = 0.2\), the approximation to the solution is \(y_{2} = 0.852967995\).

 

Now, we have:

 

$$ \begin{align*}{f_2} & =  – 0.155264954 & \hspace{0.25in}{y_3} & = 0.837441500\\ {f_3} & = 0.023922788 & \hspace{0.25in}{y_4} & = 0.839833779\\ {f_4} & = 0.1184359245 & \hspace{0.25in}{y_5}&  = 0.851677371\end{align*} $$

 

You can also double check the results using the free Euler’s method calculator.

 

Here’s a brief table with approximations and exact values for the solutions for the provided spots.

 

Time t_{n} Approximation  Exact  Error %
t_{0} = 0 y_{0} = 1 y_{0} = 1 0
t_{1} = 0.1 y_{1} = 0.9 y_{0.1} = 0.925794646 2.79
t_{2} = 0.2 y_{2} = 0.852967995 y_{0.2} = 0.889504459 4.11
t_{3} = 0.3 y_{3} = 0.837441500 y_{0.3} = 0.876191288 4.42
t_{4} = 0.4 y_{4} = 0.839833779 y_{0.4} = 0.876283777 4.16
t_{5} = 0.5 y_{5} = 0.851677371 y_{0.5} = 0.883727921 3.63

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Last Words:

In this article, we discussed the euler’s method and the use of the free euler’s method calculator in this regard.

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