# Euler Method for solving differential equation

Given a differential equation dy/dx = f(x, y) with initial condition y(x0) = y0. Find its approximate solution using Euler method.

Euler Method :
In mathematics and computational science, the Euler method (also called forward
Euler method) is a first-order numerical procedure for solving ordinary differential
equations (ODEs) with a given initial value.
Consider a differential equation dy/dx = f(x, y) with initial condition y(x0)=y0
then a successive approximation of this equation can be given by:

y(n+1) = y(n) + h * f(x(n), y(n))
where h = (x(n) – x(0)) / n
h indicates step size. Choosing smaller
values of h leads to more accurate results
and more computation time.

Example :

```    Consider below differential equation
dy/dx = (x + y + xy)
with initial condition y(0) = 1
and step size h = 0.025.
Find y(0.1).

Solution:
f(x, y) = (x + y + xy)
x0 = 0, y0 = 1, h = 0.025
Now we can calculate y1 using Euler formula
y1 = y0 + h * f(x0, y0)
y1 = 1 + 0.025 *(0 + 1 + 0 * 1)
y1 = 1.025
y(0.025) = 1.025.
Similarly we can calculate y(0.050), y(0.075), ....y(0.1).
y(0.1) = 1.11167```

Below is the implementation:

## C++

 `/* CPP  Program to find approximation` `   ``of a ordinary differential equation` `   ``using euler method.*/` `#include ` `using` `namespace` `std;`   `// Consider a differential equation` `// dy/dx=(x + y + xy)` `float` `func(``float` `x, ``float` `y)` `{` `    ``return` `(x + y + x * y);` `}`   `// Function for Euler formula` `void` `euler(``float` `x0, ``float` `y, ``float` `h, ``float` `x)` `{` `    ``float` `temp = -0;`   `    ``// Iterating till the point at which we` `    ``// need approximation` `    ``while` `(x0 < x) {` `        ``temp = y;` `        ``y = y + h * func(x0, y);` `        ``x0 = x0 + h;` `    ``}`   `    ``// Printing approximation` `    ``cout << ``"Approximate solution at x = "` `         ``<< x << ``"  is  "` `<< y << endl;` `}`   `// Driver program` `int` `main()` `{` `    ``// Initial Values` `    ``float` `x0 = 0;` `    ``float` `y0 = 1;` `    ``float` `h = 0.025;`   `    ``// Value of x at which we need approximation` `    ``float` `x = 0.1;`   `    ``euler(x0, y0, h, x);` `    ``return` `0;` `}`

## Java

 `// Java program to find approximation of an ordinary` `// differential equation using euler method` `import` `java.io.*;`   `class` `Euler {` `    ``// Consider a differential equation` `    ``// dy/dx=(x + y + xy)` `    ``float` `func(``float` `x, ``float` `y)` `    ``{` `        ``return` `(x + y + x * y);` `    ``}`   `    ``// Function for Euler formula` `    ``void` `euler(``float` `x0, ``float` `y, ``float` `h, ``float` `x)` `    ``{` `        ``float` `temp = -``0``;`   `        ``// Iterating till the point at which we` `        ``// need approximation` `        ``while` `(x0 < x) {` `            ``temp = y;` `            ``y = y + h * func(x0, y);` `            ``x0 = x0 + h;` `        ``}`   `        ``// Printing approximation` `        ``System.out.println(``"Approximate solution at x = "` `                           ``+ x + ``" is "` `+ y);` `    ``}`   `    ``// Driver program` `    ``public` `static` `void` `main(String args[]) ``throws` `IOException` `    ``{` `        ``Euler obj = ``new` `Euler();` `        ``// Initial Values` `        ``float` `x0 = ``0``;` `        ``float` `y0 = ``1``;` `        ``float` `h = ``0``.025f;`   `        ``// Value of x at which we need approximation` `        ``float` `x = ``0``.1f;`   `        ``obj.euler(x0, y0, h, x);` `    ``}` `}`   `// This code is contributed by Anshika Goyal.`

## Python3

 `# Python Code to find approximation` `# of a ordinary differential equation` `# using euler method.`   `# Consider a differential equation` `# dy / dx =(x + y + xy)` `def` `func( x, y ):` `    ``return` `(x ``+` `y ``+` `x ``*` `y)` `    `  `# Function for euler formula` `def` `euler( x0, y, h, x ):` `    ``temp ``=` `-``0`   `    ``# Iterating till the point at which we` `    ``# need approximation` `    ``while` `x0 < x:` `        ``temp ``=` `y` `        ``y ``=` `y ``+` `h ``*` `func(x0, y)` `        ``x0 ``=` `x0 ``+` `h`   `    ``# Printing approximation` `    ``print``(``"Approximate solution at x = "``, x, ``" is "``, ``"%.6f"``%` `y)` `    `  `# Driver Code` `# Initial Values` `x0 ``=` `0` `y0 ``=` `1` `h ``=` `0.025`   `# Value of x at which we need approximation` `x ``=` `0.1`   `euler(x0, y0, h, x)`

## C#

 `// C# program to find approximation of an ordinary` `// differential equation using euler method` `using` `System;`   `class` `GFG {`   `    ``// Consider a differential equation` `    ``// dy/dx=(x + y + xy)` `    ``static` `float` `func(``float` `x, ``float` `y)` `    ``{` `        ``return` `(x + y + x * y);` `    ``}`   `    ``// Function for Euler formula` `    ``static` `void` `euler(``float` `x0, ``float` `y, ``float` `h, ``float` `x)` `    ``{`   `        ``// Iterating till the point at which we` `        ``// need approximation` `        ``while` `(x0 < x) {` `            ``y = y + h * func(x0, y);` `            ``x0 = x0 + h;` `        ``}`   `        ``// Printing approximation` `        ``Console.WriteLine(``"Approximate solution at x = "` `                          ``+ x + ``" is "` `+ y);` `    ``}`   `    ``// Driver program` `    ``public` `static` `void` `Main()` `    ``{`   `        ``// Initial Values` `        ``float` `x0 = 0;` `        ``float` `y0 = 1;` `        ``float` `h = 0.025f;`   `        ``// Value of x at which we need` `        ``// approximation` `        ``float` `x = 0.1f;`   `        ``euler(x0, y0, h, x);` `    ``}` `}`   `// This code is contributed by Vt_m.`

## PHP

 ``

## Javascript

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Output

```Approximate solution at x = 0.1  is  1.11167
```

Time complexity: O(x/h)
Auxiliary space: O(1)

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