\[ \left (a^2 \sqrt {x^2+y(x)^2}-x^2\right ) y'(x)^2+a^2 \sqrt {x^2+y(x)^2}+2 x y(x) y'(x)-y(x)^2=0 \] ✓ Mathematica : cpu = 1.48383 (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 = 5.939 (sec), leaf count = 199
\[\arctan \left (\frac {x}{y \relax (x )}\right )-\frac {2 \sqrt {a^{2} \left (y \relax (x )^{2}+x^{2}\right ) \left (-a^{2}+\sqrt {y \relax (x )^{2}+x^{2}}\right )}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y \relax (x )^{2}+x^{2}}}}{a}\right )}{a \sqrt {y \relax (x )^{2}+x^{2}}\, \sqrt {-a^{2}+\sqrt {y \relax (x )^{2}+x^{2}}}}-c_{1} = 0\]