\[ \left (a^2-x^2\right ) \left (a^2-y(x)^2\right ) y''(x)+\left (a^2-x^2\right ) y(x) y'(x)^2-x \left (a^2-y(x)^2\right ) y'(x)=0 \] ✓ Mathematica : cpu = 0.40274 (sec), leaf count = 195
\[\left \{\left \{y(x)\to -\frac {1}{2} e^{-c_2} \left (\frac {a^2}{a^2-x^2}\right )^{-\frac {c_1}{2}} \sqrt {-a^2 \left (\left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{c_1}-e^{2 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{c_1}\right ){}^2}\right \},\left \{y(x)\to \frac {1}{2} e^{-c_2} \left (\frac {a^2}{a^2-x^2}\right )^{-\frac {c_1}{2}} \sqrt {-a^2 \left (\left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{c_1}-e^{2 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{c_1}\right ){}^2}\right \}\right \}\]
✓ Maple : cpu = 0.354 (sec), leaf count = 51
\[ \left \{ y \left ( x \right ) ={\frac {1}{2\,{\it \_C2}} \left ( \left ( \left ( x+\sqrt {-{a}^{2}+{x}^{2}} \right ) ^{{\it \_C1}} \right ) ^{2}{{\it \_C2}}^{2}+{a}^{2} \right ) \left ( \left ( x+\sqrt {-{a}^{2}+{x}^{2}} \right ) ^{{\it \_C1}} \right ) ^{-1}} \right \} \]