Euler’s method is considered to be one of the oldest and simplest methods to find the numerical solution of ordinary differential equation or the initial value problems. Here, a short and simple algorithm and flowchart for Euler’s method has been presented, which can be used to write program for the method in any high level programming language.

Through Euler’s method, you can find a clear expression for y in terms of a finite number of elementary functions represented with x. The initial values of y and x are known, and for these an ordinary differential equation is considered.

### What is the Euler method algorithm?

The Euler method algorithm is a numerical method which is used to solve the first order differential equation of the form y’ = f(x,y) where ‘x’ represents the independent variable and ‘y’ represents the dependent variable. The general solution of this equation is given by the formula y = y0 + y1*x. Where ‘y0’ represents the initial value of the dependent variable and ‘y1’ represents the slope of the tangent line.

Steps to get the solution of the differential equation

- Solve the homogeneous part of the equation.
- Get the particular solution of the equation.
- Substitute the value of y into the original equation and find the answer.
- Check whether the answer is the correct solution or not.
- If the answer is not the correct one, then repeat the steps 1 to 4.
- If the answer is the correct one, then stop and calculate the value of y.
- Repeat the steps 2 to 6 for all the values of x.
- Take the average of all the answers.

### Why is the Euler method algorithm the best?

- It is the best method because it is easy to implement.
- It is simple and quick to understand.
- It can be used to solve all kinds of differential equations.

**Why Use the Euler Method Algorithm?**

The Euler method algorithm is the best way to get the accurate solution of any differential equation. It is known as the simplest and easiest method to solve differential equations.

Now, lets look at the mathematics and algorithm behind the Euler’s method. A sequence of short lines is approximated to find the curve of solution; this means considering tangent line in each interval. Using the information obtained from here, the value of ‘y_{n}’ corresponding to the value of ‘x_{n}‘ is to determined by dividing the length (x_{n }– x) into n intervals or strips.

So, strip width= (x_{n }– x)/n and x_{n}=x_{0}+ nh.

Again, if m be the slope of the curve at point, y_{1= }y_{0 + m}(x_{0 }, y_{o})h.

Now, from this all the intermediate ‘y’ values can be found. This method was developed by Leonhard Euler.

## Euler’s Method Algorithm:

- Start
- Define function
- Get the values of x0, y0, h and xn

*Here x0 and y0 are the initial conditions

h is the interval

xn is the required value - n = (xn – x0)/h + 1
- Start loop from i=1 to n
- y = y0 + h*f(x0,y0)

x = x + h - Print values of y0 and x0
- Check if x < xn

If yes, assign x0 = x and y0 = y

If no, goto 9. - End loop i
- Stop

## Euler’s Method Flowchart:

Also see,

- Euler’s Method C Program
- Euler’s Method MATLAB Program
- How to Calculate Euler’s Method?
- The Euler’s Method To Calculate Integrals
- How To Solve A Linear Equation Using Euler’s Method

Euler’s method is very simple and easy to understand and use in programming to solve initial value problems. You can refer the aforementioned algorithm and flowchart to write a program for Euler’s method in any high level programming language.

**Note**: Euler’s method cannot be regarded as one of the best approaches to solve ordinary differential equations. It is very slow, but its modified form, Modified Euler’s Method is fast.

Thank you!