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Euler method is used to solve the differential equation. Euler method yields solutions in the form of numerical values of y for discrete values of x at specified intervals. It is based on the idea of linear approximation in which small tangent lines on a short distance are used to get the approximate solution to the initial-value problem.

DERIVATION

In order to solve the first-order differential equation:

The solution starts from the initial value $\mathrm{y} = {\mathrm{y}}_0$ at $\mathrm{x} = {\mathrm{x}}_0$ and in the first step it is taken as $\mathrm{x} = {\mathrm{x}}_0+ \mathrm{h}$ and then the value of ${\mathrm{y}}_1$ of the solution is computed. Using this value of ${\mathrm{y}}_1$the next value ${\mathrm{y}}_2$ is computed at $\mathrm{x} = {\mathrm{x}}_0+ 2\mathrm{h}$. 

In a similar way, the value of y is carried out at the discrete values of x 

Each step of the Euler method is performed using a formula which is derived from Taylor’s series as follows:

Now we take p = 1, which gives 

Therefore,

So, in general, we can say that:

These successive values are calculated using recursion relation.

GRAPHICAL REPRESENTATION

INPUT: 

A function f(x)

OUTPUT: 

The value after differentiating f(x)

PROCESS:

Step 1: [Defining f(x,y)]
	return x × y

Step 2: [Euler Method]
	Read the values of x0, y0 i.e. the initial values to calculate the differentiation.
	Read h(the step length)
	Read xn(the ending value to terminate the calculation)
	While x0 ≤ xn repeat
		Set y1 ← y0 + h × f(x0,y0)
		Print x0, y0
		Set x0 = x0 + h
		Set y0 = y1
	[End of ‘while’]
[End of ‘Euler’ method]