Determining which recursive term is bigger if they share the same definition
Let $c_3 = \max(1,x,y,c_1+c_2)$. Then
$$
a_1 = x \stackrel{c_3 \geq x}\leq c_3^1
$$
and
$$
a_2 = y \stackrel{c_3 \geq y}\leq c_3 \stackrel{c_3 \geq 1}\leq c_3^2.
$$
Now suppose that $a_{n-2} \leq c_3^{n-2}$ and $a_{n-1} \leq c_3^{n-1}$. Then
$$
a_n = c_1 a_{n-1} + c_2 a_{n-2} \stackrel{\text{assumption}}\leq c_1 c_3^{n-1} + c_2 c_3^{n-2} \stackrel{c_3 \geq 1}\leq c_1 c_3^{n-1} + c_2 c_3^{n-1} \stackrel{c_3 \geq c_1+c_2}\leq c_3^n.
$$