In this blog, I will be discussing about Picard method of solving the ordinary differential equation. 



Discuss the Method of Solving differntial equation y' = f(x,y) ,  \(y = y_0\) for \( x = x_0\) by Picard Method.

Solution:

Given differential equation is, 

$$ y' = f(x,y) $$

Integrating both side,

\(\int_{x_0}^{x_1}y'.dx = \int_{x_0}^{x_1}f(x,y)dx \)

or, \([y']_{x_0}^{x_1} = \int_{x_0}^{x_1}f(x,y)dx\)

or, \(y(x_1) - y(x_0) = \int_{x_0}^{x_1}f(x,y)dx\)

or, \(y(x_1) = y(x_0)+\int_{x_0}^{x_1}f(x,y)dx\)

Integral \(\int_{x_0}^{x_1}f(x,y)dx\) cannot be evaluated directly as f(x,y) contains y.

According to Picard, Replace y by \(y(x_0)\), then 

\(y(x_1) = y(x_0)+\int_{x_0}^{x_1}f(x,y(x_0))dx\)

To find the Iteration Formula, 

Replacing \(x_1 \) with  \(x_{i+1} \) and \(x_0\)  with \(x_i \) we get or required formula.

$$y(x_{i+1}) = y(x_i)+\int_{x_i}^{x_{i+1}}f(x,y(x_i))dx$$