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This method is used to solve the first-order differential equation. This is one of the earliest analytic numeric algorithms which can give an approximate solution of initial value problems for ordinary differential equations. This algorithm is not applied frequently.

DERIVATION

Let consider the first-order differential equation 

Differentiating the equation with respect to x, we get

Differentiating successively we can obtain y”, y’’’, ...

We get y(x) for all values of x the above equation converges. 

Let, 

And let

And we get

In this way, we get a discrete set of values ${\mathbf{y}}_{\mathbf{n}}$ which are the approximations to the actual values of y at the points 

INPUT: 

A function f(x)

OUTPUT: 

The value after performing the differentiation

PROCESS: 

Step 1: [defining a function f(x)]
Return exponential(x)/(x^3 - 1)
[End of function ‘f(x)’]

Step 2: [Taylor’s Method]
             Set a ← 2.5
		Set b ← 4.5
		for k = 0 to 3 repeat
			Set t1 ← 0
			Set h ← (b-a)/2^k
			Set n ← 2^k
			Set q ← 0
			Set e ← f(a)
			Print h
			Print a
			Print b
			Print q, a, e
			for i = 1 to n - 1 repeat
				Set x1 ← a + h × i
				Set c ← f(x1)
				Set t1 ← t1 + c
				Print i, x1, c
			[End of ‘for’ loop]
			Set g ← f(b)
			Print n, b, g
			Set l[k] ← h × ((e/2) + t1 + (g/2))
			Set d[0][k] ← l[k]
			Print l[k]
		[End of ‘for’ loop]
		Set p ← 2
		for s = 1 to 3 repeat
			for r = 0 to p repeat
				Set d[s][r] ← d[s-1][r+1]+(d[s-1][r+1]-d[s-1][r])/(2^2s-1)
[End of ‘for’ loop]
			Set p ← p - 1
		[End of ‘for’ loop]
		Print d[0][0]
		Print d[0][1], d[1][0]
		Print d[0][2], d[1][1], d[2][0]
		Print d[0][3], d[1][2], d[2][1], d[3][0]
[End of Taylor’s method]