ODE
\[ \left (a^2-x^2\right ) \left (a^2-y(x)^2\right ) y''(x)+\left (a^2-x^2\right ) y(x) y'(x)^2=x \left (a^2-y(x)^2\right ) y'(x) \] ODE Classification
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.321694 (sec), leaf count = 65
\[\left \{\left \{y(x)\to \frac {1}{2} e^{c_2} \left (\sqrt {x^2-a^2}+x\right )^{-c_1}+\frac {1}{2} a^2 e^{-c_2} \left (\sqrt {x^2-a^2}+x\right )^{c_1}\right \}\right \}\]
Maple ✓
cpu = 0.034 (sec), leaf count = 40
\[ \left \{ \ln \left ( y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2}} \right ) -{\it \_C1}\,\ln \left ( x+\sqrt {-{a}^{2}+{x}^{2}} \right ) -{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[(a^2 - x^2)*y[x]*y'[x]^2 + (a^2 - x^2)*(a^2 - y[x]^2)*y''[x] == x*(a^2 - y[x]^2)*y'[x],y[x],x]
Mathematica raw output
{{y[x] -> E^C[2]/(2*(x + Sqrt[-a^2 + x^2])^C[1]) + (a^2*(x + Sqrt[-a^2 + x^2])^C[1])/(2*E^C[2])}}
Maple raw input
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), y(x),'implicit')
Maple raw output
ln(y(x)+(y(x)^2-a^2)^(1/2))-_C1*ln(x+(-a^2+x^2)^(1/2))-_C2 = 0