## Solving for \$y” – 4y’ – 5y – 2 = 0\$

I am looking to solve for the above nonhomogeneous ODE. I know how to find the general solution for the reduced equation of the homogeneous form, that is, $$y” – 4y’ – 5y = 0.$$

The characteristic equation is $$r^{2} – 4r – 5 = 0$$, which gives two real and distinct roots $$r=-1,5$$.

So the complementary solution is $$y_{c}=c_{1}e^{5x} + c_{2}e^{-x}$$.

Now I am looking to guess the particular on the right-hand side but I am not sure about how to do that in order to find the general solution of the above nonhomogenous ODE.