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This method is used to solve the differential equation in numerical. It is a family of the implicit and explicit iterative method. It is very simple to program and their truncation error can be controlled in a more straightforward manner than for multiple methods. The Runge Kutta method were the usual tool for calculating the needed initial values for the multistep method.

The method propagates a solution over an interval by combining different Euler-style steps. Each step is evaluated with different parameters.

DERIVATION

Suppose the differential equation be represented as:

Where

And therefore, the recursion is given by

In an s stage Runge kutta method, 

and 

The constants ${\mathrm{\alpha }}_{\mathrm{j}}$, ${\mathrm{\gamma }}_{\mathrm{j}}$ and ${\mathrm{w}}_{\mathrm{j}}$ are to be determined such that 

For the fourth-order Runge Kutta method we consider:

Now by determining the values of the constants we get:

Where

GRAPHICAL REPRESENTATION

INPUT: 

A function f(x, y)

OUTPUT: 

The value after solving the differential equation

PROCESS:

Step 1: [defining function f(x,y)]
	Set a ← 1 and b ← 3
	Return f(x, y)

Step 2: [defining runge kutta method]
	Set x[0] ← 0.0
	Set y[0] ← 1.0
	Set f[0] ← g(0.0,1.0)
	Set h ← 0.1
	for j = 0 to 9 repeat
		Set f[j] ← g(x[j], y[j])
		Set k1 ← h × g(x[j], y[j])
		Set k2 ← h × g(x[j] + h/2.0, y[j] + k1/2.0)
		Set k3 ← h × g(x[j] + h/2.0, y[j] + k2/2.0)
		Set k4 ← h × g(x[j] + h, y[j] + k3)
		Set y[j + 1] ← y[j] + ((k1 + k4 + 2.0 × (k2 + k3))/6.0)
		Set x[j + 1] ← x[j] + h
		Print x[j], y[j], f[j]
	[End of ‘for’ loop]
	Print y[9]
[End of runge kutta method]