Quasi-linear utility. Deriving demand

I was trying to derive a general demand for the good $x$ for this quasi-linear function $u(x,y) = y + 2sqrt{x}$ subject to standard budget constraint $p_x x + p_y y leq M$

Using Kuhn-Tucker conditions (my aim was to account the corner solutions) I obtained the following results:

$x = 0$, $y = M / p_y$ when $MRS < p_x / p_y$

$x = M / p_x$, $y = 0$ when $MRS > p_x / p_y$

$x = (p_y / p_x) ^ 2, y = M / p_y – p_y / p_x$ when $MRS = p_x / p_y$

The problem is that I do not how to write down the demand for $x$ with all conditions listed above ($MRSlesseqgtr$ ) in an explicit form. In other words, I think it is possible to be left only with prices and income as conditions for the demand.

I hope that I was clear