Min-Max recurrence relations

Question

Now here I am not getting any idea whether it's linear homogeneous or linear non-homogeneous equation. How can this equation be solved correctly?

Answer 1

Hint:

Play with the first few $T(n)$'s and you'll discover that everything depends on a simple relationship between $T(1)$ and $T(2)$ in a consistent way.

! If $2x+2>y$, the first few $T(n)$'s are: $$x, y, x+y+2, 2y+2, x+2y+4, 3y+4, $$ If $2x+2\leqslant y$, the first few $T(n)$'s are: $$ x, y, x+y+2, 2x+y+4, 3x+y+6,4x+y+8,$$ Do you see the pattern?