Consider the equation

$$\frac{d y}{d x}=x\left(\frac{1-y^{2}}{1-x^{2}}\right)^{1 / 2}$$

where the positive square root is taken, within the square $\mathcal{S}: 0 \leqslant x<1,0 \leqslant y \leqslant 1$. Find the solution that begins at $x=y=0$. Sketch the corresponding solution curve, commenting on how its tangent behaves near each extremity. By inspection of the righthand side of $(*)$, or otherwise, roughly sketch, using small line segments, the directions of flow throughout the square $\mathcal{S}$.