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Picard’s iterative method helps to solve the differential equation by a sequence of approximations as ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{,}\mathbf{ }{\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{,}\mathbf{ }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{ }{\mathbf{y}}_{\mathbf{k}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ to the solution in which the nth approximation depends on the previous approximations. 

It is an iterative method used for approximating the solutions of differential equations. The solution of the differential equation will be more accurate if it is used repeatedly. It is easier to implement this method and the solution generally in power series.

DERIVATION

If we consider the first-order differential equation as:

Integrating between limits, we get

Now, by replacing $\mathrm{y} = {\mathrm{y}}_0$ as

The second approximation is:

Continuing the process, the nth approximation is:

This is known as recursion or iterative formula.

Now, this Picard method generates a sequence of approximations as ${\mathrm{y}}_1, {\mathrm{y}}_2, {\mathrm{y}}_3, ....$ which converges to the exact solution y(x). So the function f(x,y) is bounded in the neighborhood of the point $({\mathrm{x}}_0, {\mathrm{y}}_0)$ and it satisfies the Lipschitz conditions.

GRAPHICAL REPRESENTATION

INPUT: 

A function f(x)

OUTPUT: 

The solution after performing picard’s method

PROCESS:

Step 1: [defining y1]
	return (1 + (x) + x^2/2)

Step 2: [defining y2]
	return (1 + (x) + x^2/2 + x^3/3 + x^4/8)

Step 3: [defining y3]
	return (1 + (x) + x^2/2 + x^3/3 + x^4/8 + x^5/15 + x^6/48)

step 4:[Picard’s method]
	read x0, xn [the lower and upper limit of the integration] and the allowed error
	      Set c ← 0
	      for tmp = x0 to xn repeat 
		Set y1[c] ← f1(tmp)
		Set y2[c] ← f2(tmp)
		Set y3[c] ← f3(tmp)
		Set tmp ← tmp + err
		Set c ← c + 1
	     [End of ‘for’ loop]
[printing the value of x]
for tmp = x0 to xn repeat
	Print tmp
	Set tmp ← tmp + err	
      [End of ‘for’]
[printing the values of y1]
    Set c ← 0
for tmp = x0 to xn repeat 
	print y1(c)
	Set tmp ← tmp + err
	Set c ← c + 1
[End of ‘for’ loop]
    [printing the values of y2]
    Set c ← 0
for tmp = x0 to xn repeat 
	print y2(c)
	Set tmp ← tmp + err
	Set c ← c + 1
[End of ‘for’ loop]
    [printing the values of y3]
    Set c ← 0
for tmp = x0 to xn repeat 
	print y3(c)
	Set tmp ← tmp + err
	Set c ← c + 1
[End of ‘for’ loop]