Solving recurrence relations of type $S_n-7S_{n-1}+10S_{n-2}=5\cdot 3^n$ I know how to solve this kind of equasions 
$$S_n-7S_{n-1}+10S_{n-2}=5\cdot 3^n$$ $$S_0=0, S_1=1$$ for example...but when there is a constant (example:$(3^n+5))$ or  $(5\cdot 3^n)$ i don't know how to solve it. Any tips.

Since the characteristic polynomial is $z^2-7z+10 = (z-2)(z-5)$, the solutions of
$$ S_{n}-7 S_{n-1} + 10 S_{n-2} = 0 $$
have the form $S_n = \alpha 2^n+\beta 5^n$. By direct inspection a solution of
$$ S_{n}-7 S_{n-1} + 10 S_{n-2} = 5\cdot 3^n $$
is given by $S_n=-\frac{45}{2}\cdot3^n$, hence the set of solutions of the previous recurrence is given by
$$ S_n = \alpha 2^n+\beta 5^n-\frac{45}{2}3^n $$
and by imposing $S_0=0$ and $S_1=1$ we get:
$$\boxed{ S_n = \color{red}{\frac{44}{3}2^n + \frac{47}{6} 5^n -\frac{45}{2} 3^n}.} $$
