We start by solving $r^3-r=0$ which gives that $r_1=0, r_2=1, r_3=-1$.

Variation of parameters is not a good method to guess a particular solution here. You can try guessing that the particular solution is $Y_p=A\sin(t)+B\cos(t)$ or just look at the equation and deduce that $Y_p=\cos(t)$

So, general solution to the equation is

$Y_G=\cos(t)+c_1+c_2e^t+c_3e^{-t}$.