Hello,

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.

First recurrence:

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!