Differential Equations | Part IA, 2002

Consider the equation

dydx=x(1y21x2)1/2\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 S:0x<1,0y1\mathcal{S}: 0 \leqslant x<1,0 \leqslant y \leqslant 1. Find the solution that begins at x=y=0x=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 S\mathcal{S}.

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