Hey all, i gonna share another problem to solve here we go:

Consider the equation

$\displaystyle y''+y=f(x)$

where

$\displaystyle y(0)=X_0 ; y'(0)=V_0$

prove that the solution is:

$\displaystyle y(x)=\int_{0}^{x}sin(x-s)f(s)ds + X_0cosx+V_0senx$