Please provide a complete working Python code. The max error did not decrease in the previous solution provided. The approx solution should get closer to exact solution with increasing values of n$.$ Chatgpt gives wrong answers and do not print false outputs which look correct.

Consider the following second order ODE:

y$+2$y$+$ y $=0,$

with y$(0)=2$ and y$(0)=1,$ for $0<=$ x $<=1.$

Write the finite difference discretization of this ODE using central finite difference approximations for both y$($x$)$ and y$($x$).$ Use it to approximate the exact solution y$($x$)=$

$2$e$^$x $+$ xe$^$x with n $=2^$i intervals, i $=3,...,6.$ Discuss the convergence of your finite

difference discretization scheme