Deriving demand function from utility function
This is an example of quasilinear utility. In general, a quasilinear utility function takes the form 
$$u(x,y)=f(x)+y.$$
The perfect substitutes utility function, for example, is a special case of the quasilinear utility function. One feature of quasilinear utility is that the MRS is independent of at least one of the two goods:
$$\frac{\frac{\partial u(x,y)}{\partial x}}{\frac{\partial u(x,y)}{\partial y}}=f'(x),$$
meaning the slope of indifference curves is independent of the quantity of good $y$. 

The Marshallian demand functions satisfy the equations:
$$ f'(x)=\frac{P_x}{P_y}$$
$$I=P_xx+P_yy,$$

which come from the first-order conditions of the constrained maximization problem. We can solve for the Marshallian demand function for $x$ directly from the first equation:
$$x^*=f'^{-1}(\frac{P_x}{P_y}).$$
Substituting this into your second equation gives 
$$I=P_xf'^{-1}(\frac{P_x}{P_y})+P_yy$$
$$y^*=\frac{I-P_xf'^{-1}(\frac{P_x}{P_y})}{P_y}.$$

In your example, we get
$$x^*=(\frac{P_y}{P_x})^2.$$
$$y^*=\frac{I-P_x(\frac{P_y}{P_x})^2}{P_y}.$$

There is a fundamental problem with your solution. You write a demand function for $x$ that depends on the quantity of $y$. Demand functions depend either on $(P_x,P_y,I)$, if Marshallian, or $(P_x,P_y,U)$, if Hicksian.