I have two textbook examples of simple non-homogeneous recurrences with model solutions, but there's one thing that I don't understand in both of them.
Characteristic equation ( ) has double characteristic root 2.
Non-homogeneous part is first degree polynomial, so the guess is . Then we substitute it to original recurrence and get following equation:
Now comes the part, that I don't understand. Textbook says, that we get two solutions ( and ) from that equation. But it completely baffles me, how I'am supposed to get those solutions. I think it have something to do with method of undetermined coefficients, but we haven't got taught anything about it, gg. Rest of the solution I understand.
Second recurrence: (no other information given)
Guess is form . And subtitution to recurrence:
Now I don't understand even, how that right side of biconditional is got.
When I reduce that equation, I get something like: . Again equation is "solved" and and is got, which I don't understand at all.
So, any help is appreciated. Thank you!