\[ \boxed { \left ( \left ( y \left ( x \right ) \right ) ^{2}+ \left ( 1-a \right ) {x}^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+2\,axy \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( 1-a \right ) \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}=0} \]
Mathematica: cpu = 0.110014 (sec), leaf count = 83 \[ \left \{\text {Solve}\left [\sqrt {a-1} \tan ^{-1}\left (\frac {y(x)}{x}\right )-\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=c_1+\log (x),y(x)\right ],\text {Solve}\left [\sqrt {a-1} \tan ^{-1}\left (\frac {y(x)}{x}\right )+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )=c_1-\log (x),y(x)\right ]\right \} \]
Maple: cpu = 0.905 (sec), leaf count = 61 \[ \left \{ y \left ( x \right ) =\tan \left ( {\it RootOf} \left ( -2\,{\it \_Z}\,\sqrt {a-1}-\ln \left ( {\frac {{x}^{2}}{ \left ( \cos \left ( { \it \_Z} \right ) \right ) ^{2}}} \right ) +2\,{\it \_C1} \right ) \right ) x,y \left ( x \right ) =\tan \left ( {\it RootOf} \left ( 2\,{ \it \_Z}\,\sqrt {a-1}-\ln \left ( {\frac {{x}^{2}}{ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2}}} \right ) +2\,{\it \_C1} \right ) \right ) x \right \} \]