\[ \left (a^2-x^2\right ) \left (a^2-y(x)^2\right ) y''(x)+\left (a^2-x^2\right ) y(x) y'(x)^2-x \left (a^2-y(x)^2\right ) y'(x)=0 \] ✓ Mathematica : cpu = 0.309609 (sec), leaf count = 363
\[\left \{\left \{y(x)\to -\frac {1}{2} e^{-c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{-\frac {c_1}{2}} \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{-\frac {c_1}{2}} \sqrt {2 a^2 e^{2 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{c_1} \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{c_1}-a^2 \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{2 c_1}-a^2 e^{4 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{2 c_1}}\right \},\left \{y(x)\to \frac {1}{2} e^{-c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{-\frac {c_1}{2}} \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{-\frac {c_1}{2}} \sqrt {2 a^2 e^{2 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{c_1} \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{c_1}-a^2 \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{2 c_1}-a^2 e^{4 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{2 c_1}}\right \}\right \}\] ✓ Maple : cpu = 1.62 (sec), leaf count = 51
\[\left \{y \left (x \right ) = \frac {\left (c_{2}^{2} \left (x +\sqrt {-a^{2}+x^{2}}\right )^{2 c_{1}}+a^{2}\right ) \left (x +\sqrt {-a^{2}+x^{2}}\right )^{-c_{1}}}{2 c_{2}}\right \}\]