\[ \left (a^2-x^2\right ) \left (a^2-y(x)^2\right ) y''(x)+\left (a^2-x^2\right ) y(x) y'(x)^2-x \left (a^2-y(x)^2\right ) y'(x)=0 \] ✓ Mathematica : cpu = 0.250782 (sec), leaf count = 363
DSolve[-(x*(a^2 - y[x]^2)*Derivative[1][y][x]) + (a^2 - x^2)*y[x]*Derivative[1][y][x]^2 + (a^2 - x^2)*(a^2 - y[x]^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {1}{2} e^{-c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{-\frac {c_1}{2}} \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{-\frac {c_1}{2}} \sqrt {2 a^2 e^{2 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{c_1} \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{c_1}-a^2 \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{2 c_1}-a^2 e^{4 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{2 c_1}}\right \},\left \{y(x)\to \frac {1}{2} e^{-c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{-\frac {c_1}{2}} \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{-\frac {c_1}{2}} \sqrt {2 a^2 e^{2 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{c_1} \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{c_1}-a^2 \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )^{2 c_1}-a^2 e^{4 c_2} \left (1-\frac {x}{\sqrt {x^2-a^2}}\right )^{2 c_1}}\right \}\right \}\] ✓ Maple : cpu = 0.209 (sec), leaf count = 51
dsolve((a^2-x^2)*(a^2-y(x)^2)*diff(diff(y(x),x),x)+(a^2-x^2)*y(x)*diff(y(x),x)^2-x*(a^2-y(x)^2)*diff(y(x),x)=0,y(x))
\[y \left (x \right ) = \frac {\left (\left (x +\sqrt {-a^{2}+x^{2}}\right )^{2 c_{1}} c_{2}^{2}+a^{2}\right ) \left (x +\sqrt {-a^{2}+x^{2}}\right )^{-c_{1}}}{2 c_{2}}\]