I have recurrence relation
$\displaystyle x_k- x_{k-1}= 7^k$
Any hints how I can take out the $\displaystyle 7^k$ term?
Well, yes, just drop it! That is, the "associated homogeneous relation" is $\displaystyle x_k- x_{k-1}= 0$. Because the original recurrence relation is linear, its general solution is the sum of the general solution to the associated homogeneous relation and any one solution to the entire relation.
I'm not sure I follow, could you show me how to go further with that?
I take $\displaystyle x_k - x_{k-1 }= 0$
Which gives me $\displaystyle x^2-x = 0$
Which gives x = 0, x = 1
Which would make for a general solution containing $\displaystyle 0^k$ and $\displaystyle 1^k$..