\[ \boxed { \left ( {a}^{2}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}-{x}^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+2\,xy \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +{a}^{2}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}- \left ( y \left ( x \right ) \right ) ^{2}=0} \]
Mathematica: cpu = 1.555697 (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 = 8.112 (sec), leaf count = 199 \[ \left \{ \arctan \left ( {\frac {x}{y \left ( x \right ) }} \right ) -2\,{ \frac {\sqrt {{a}^{2} \left ( \left ( y \left ( x \right ) \right ) ^{2}+ {x}^{2} \right ) \left ( -{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}} \right ) }}{a\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}\sqrt {-{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}}\arctan \left ( {\frac {\sqrt {-{a}^{2 }+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}}{a}} \right ) }-{\it \_C1}=0,\arctan \left ( {\frac {x}{y \left ( x \right ) } } \right ) +2\,{\frac {\sqrt {{a}^{2} \left ( \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2} \right ) \left ( -{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}} \right ) }}{a\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}\sqrt {-{a}^{2}+ \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}}\arctan \left ( {\frac {\sqrt {-{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}}{a}} \right ) }-{\it \_C1}=0 \right \} \]