\[ \left ((1-a) x^2+y(x)^2\right ) y'(x)^2+2 a x y(x) y'(x)+(1-a) y(x)^2+x^2=0 \] ✓ Mathematica : cpu = 0.21179 (sec), leaf count = 83
\[\left \{\text {Solve}\left [\sqrt {a-1} \tan ^{-1}\left (\frac {y(x)}{x}\right )-\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=\log (x)+c_1,y(x)\right ],\text {Solve}\left [\sqrt {a-1} \tan ^{-1}\left (\frac {y(x)}{x}\right )+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=-\log (x)+c_1,y(x)\right ]\right \}\] ✓ Maple : cpu = 0.881 (sec), leaf count = 61
\[y \relax (x ) = \tan \left (\RootOf \left (-2 \textit {\_Z} \sqrt {a -1}-\ln \left (\frac {x^{2}}{\cos \left (\textit {\_Z} \right )^{2}}\right )+2 c_{1}\right )\right ) x\]