\[ \boxed { a{x}^{2} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-2\,axy \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}-a \left ( a-1 \right ) {x}^{2}=0} \]
Mathematica: cpu = 0.154520 (sec), leaf count = 118 \[ \left \{\left \{y(x)\to \frac {1}{2} e^{-c_1} x^{1-\frac {\sqrt {a-1}}{\sqrt {a}}} \left (e^{2 c_1}-a x^{\frac {2 \sqrt {a-1}}{\sqrt {a}}}\right )\right \},\left \{y(x)\to \frac {1}{2} e^{-c_1} x^{1-\frac {\sqrt {a-1}}{\sqrt {a}}} \left (e^{2 c_1} x^{\frac {2 \sqrt {a-1}}{\sqrt {a}}}-a\right )\right \}\right \} \]
Maple: cpu = 0.593 (sec), leaf count = 138 \[ \left \{ y \left ( x \right ) =\sqrt {-a}x,y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) -\int ^{{\it \_Z}}\!{\frac {1}{{ {\it \_a}}^{2}a-{{\it \_a}}^{2}+{a}^{2}-a}\sqrt { \left ( {{\it \_a}}^{ 2}a-{{\it \_a}}^{2}+{a}^{2}-a \right ) a}}{d{\it \_a}}+{\it \_C1} \right ) x,y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) +\int ^{{\it \_Z}}\!{\frac {1}{{{\it \_a}}^{2}a-{{\it \_a}}^{ 2}+{a}^{2}-a}\sqrt { \left ( {{\it \_a}}^{2}a-{{\it \_a}}^{2}+{a}^{2}-a \right ) a}}{d{\it \_a}}+{\it \_C1} \right ) x,y \left ( x \right ) =- \sqrt {-a}x \right \} \]