\[ \left (a^2 \sqrt {x^2+y(x)^2}-x^2\right ) y'(x)^2+a^2 \sqrt {x^2+y(x)^2}+2 x y(x) y'(x)-y(x)^2=0 \] ✓ Mathematica : cpu = 1.79877 (sec), leaf count = 229
\[\left \{\text {Solve}\left [\tan ^{-1}\left (\frac {x}{y(x)}\right )-\frac {2 \sqrt {a^2 \left (x^2+y(x)^2\right ) \left (\sqrt {x^2+y(x)^2}-a^2\right )} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+y(x)^2}-a^2}}{a}\right )}{a \sqrt {x^2+y(x)^2} \sqrt {\sqrt {x^2+y(x)^2}-a^2}}=c_1,y(x)\right ],\text {Solve}\left [\frac {2 \sqrt {a^2 \left (x^2+y(x)^2\right ) \left (\sqrt {x^2+y(x)^2}-a^2\right )} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+y(x)^2}-a^2}}{a}\right )}{a \sqrt {x^2+y(x)^2} \sqrt {\sqrt {x^2+y(x)^2}-a^2}}+\tan ^{-1}\left (\frac {x}{y(x)}\right )=c_1,y(x)\right ]\right \}\] ✓ Maple : cpu = 5.067 (sec), leaf count = 199
\[\left \{-c_{1}+\arctan \left (\frac {x}{y \left (x \right )}\right )-\frac {2 \sqrt {\left (x^{2}+y \left (x \right )^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}\right ) a^{2}}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}}}{a}\right )}{\sqrt {x^{2}+y \left (x \right )^{2}}\, \sqrt {-a^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}}\, a} = 0, -c_{1}+\arctan \left (\frac {x}{y \left (x \right )}\right )+\frac {2 \sqrt {\left (x^{2}+y \left (x \right )^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}\right ) a^{2}}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}}}{a}\right )}{\sqrt {x^{2}+y \left (x \right )^{2}}\, \sqrt {-a^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}}\, a} = 0\right \}\]