Solve the following recurrence relations using generating functions.

$($a$)a_k=3a_{k-1}+5(7^k),k1$ and $a_0=2.$

$($b$)a_k-a_{k-1}=k+1,k1,a_0=1.$

$($c$)a_{k+2}-2a_{k+1}+a_k=2^k,k0,a_0=1$ and $a_1=2.$