I have this differential equation:
$$\ddot{x}-\frac{1}{6} \dot{x} - \frac{1}{6}x = e^t$$
When I DSolve it looks like this:
DSolve[x''[t] - 1/6 x'[t] - 1/6 x[t] == Exp[t], x[t], t]
Which gives me the correct answer. With that, we know I have the characteristic polynomial:
z^2 - 1/6 z - 1/6
I also find the roots:
Root[%,1]
Root[%%,2]
So all the results are correct. My question is, how can make this more easy for me, so I don't need to type manually the characteristic polynomial with its roots? I have tried with the function StringReplace to rewrite my differential equation into the characteristic polynomial, but it was messy and not functioning properly.