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]
```

__ADVANTAGES__

**1.** It is very useful for derivations.

**2.** This method can be used to get theoretical error bounds.

**3.** Power series can be inverted to develop the inverse function.

**4.** Object reference model parameters are embedded as variables in this method.

__DISADVANTAGES__

**1.** In this method the successive terms get very complex and become difficult to derive.

**2.** Truncation error tends to grow rapidly which is away from the expansion point.

**3.** It is always not as efficient as curve fitting or direct approximation.

__APPLICATIONS__

**1.** This method is used to calculate the value of an entire function at each point while all of its derivatives are known at a single point.

**2.** The partial sums can be used as approximations of the function.

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