Consider the second$-$order differential equation $y^{\text{'}\text{'}}-3y^{\text{'}}+2y=e^x+e^{2x}+e^{-x}$

$(1)$Find the general solution of the homogeneous ordinary differential equation$($ODE$).$

$(2)$Find the particular solution.

$(3)$Find the general solution of the non$-$homogeneous ODE.