\[ \boxed { \left ( a\sqrt { \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1}-x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-1=0} \]
Mathematica: cpu = 0.352545 (sec), leaf count = 347 \[ \left \{\left \{y(x)\to \frac {-2 \sqrt {x^2 \left (a^2+c_1^2-x^2\right )}+c_1 x \log \left (-c_1 \left (\sqrt {x^2 \left (a^2+c_1^2-x^2\right )}+c_1 x\right )+a^2 (-x)+a x^2\right )+c_1 x \log \left (c_1 \left (\sqrt {x^2 \left (a^2+c_1^2-x^2\right )}+c_1 x\right )+a^2 x+a x^2\right )+c_1 x \log (x-a)-c_1 x \log (x (x-a))-c_1 x \log (a+x)-c_1 x \log (x (a+x))}{2 x}+c_2\right \},\left \{y(x)\to c_2-\frac {-2 \sqrt {x^2 \left (a^2+c_1^2-x^2\right )}+c_1 x \log \left (-c_1 \left (\sqrt {x^2 \left (a^2+c_1^2-x^2\right )}+c_1 x\right )+a^2 (-x)+a x^2\right )+c_1 x \log \left (c_1 \left (\sqrt {x^2 \left (a^2+c_1^2-x^2\right )}+c_1 x\right )+a^2 x+a x^2\right )+c_1 (-x) \log (x-a)-c_1 x \log (x (x-a))+c_1 x \log (a+x)-c_1 x \log (x (a+x))}{2 x}\right \}\right \} \]
Maple: cpu = 0.717 (sec), leaf count = 99 \[ \left \{ y \left ( x \right ) =\int \!{\frac {1}{a \left ( {a}^{2}-{x}^{2 } \right ) } \left ( -{\it \_C1}\,{a}^{2}+x\sqrt {{a}^{2} \left ( {{\it \_C1}}^{2}+{a}^{2}-{x}^{2} \right ) } \right ) }\,{\rm d}x+{\it \_C2},y \left ( x \right ) =\int \!-{\frac {1}{a \left ( {a}^{2}-{x}^{2} \right ) } \left ( {\it \_C1}\,{a}^{2}+x\sqrt {{a}^{2} \left ( {{\it \_C1}}^{2}+{a}^{2}-{x}^{2} \right ) } \right ) }\,{\rm d}x+{\it \_C2} \right \} \]