Oh okay, sorry for the confusion. The characteristic polynomial comes from assuming a solution, i.e., we assume and plug this into the originial DEQ. Thus we obtain the following
which leads us to
This tells us the characteristic equation is
When we find the roots to this equation it tells us that we have one solution containing and another solution containing , i.e., we have one solution and a second solution However, since both these are solutions to the differential equation we know that a linear combination of the two is also a solution to the differential equation, and thus we obtain the general solution
I really hope I finally answered your question. I suppose after all this you will probably understand the problem more thoroughly.
Your absolutely correct, I made a mistake and have corrected the post. We assume this is the solution, because we want
This will only occur if have the same form. Well, we know that the derivative of is itself, so this clues us into the type of solution we need, i.e., .