This is a homogeneous DE, you don't need to find a particular solution.
Just substitute your boundary conditions, noting that $\displaystyle \displaystyle y' = -3c_1e^{-3x} + \frac{c_2}{3}e^{\frac{x}{3}}$, to evaluate $\displaystyle \displaystyle c_1$ and $\displaystyle \displaystyle c_2$.