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 = 0.416688 (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 = 1.028 (sec), leaf count = 51
\[\left [y \left (x \right ) = \frac {\left (\left (x +\sqrt {-a^{2}+x^{2}}\right )^{2 \textit {\_C1}} \textit {\_C2}^{2}+a^{2}\right ) \left (x +\sqrt {-a^{2}+x^{2}}\right )^{-\textit {\_C1}}}{2 \textit {\_C2}}\right ]\] 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] -> -1/2*Sqrt[-(a^2*(-(E^(2*C[2])*(1 - x/Sqrt[-a^2 + x^2])^C[1]) + (1 + x/Sqrt[-a^2 + x^2])^C[1])^2)]/(E^C[2]*(a^2/(a^2 - x^2))^(C[1]/2))}, {y[x] -> Sqrt[-(a^2*(-(E^(2*C[2])*(1 - x/Sqrt[-a^2 + x^2])^C[1]) + (1 + x/Sqrt[-a^2 + x^2])^C[1])^2)]/(2*E^C[2]*(a^2/(a^2 - x^2))^(C[1]/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))
Maple raw output
[y(x) = 1/2*(((x+(-a^2+x^2)^(1/2))^_C1)^2*_C2^2+a^2)/((x+(-a^2+x^2)^(1/2))^_C1)/_C2]