\[ \boxed { \left ( {a}^{2}-1 \right ) {x}^{2} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+2\,xy \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2}+{a}^{2}{x}^{2}=0} \]
Mathematica: cpu = 0.573073 (sec), leaf count = 395 \[ \left \{\text {Solve}\left [\frac {a \left (-\log \left (\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}-i\right )}\right )+\log \left (-\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2+\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}+i\right )}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )-2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )}{2 \left (a^2-1\right )}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ],\text {Solve}\left [\frac {2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \left (\log \left (-\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}-i\right )}\right )-\log \left (\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2+\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}+i\right )}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )}{2 \left (a^2-1\right )}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ]\right \} \]
Maple: cpu = 1.170 (sec), leaf count = 229 \[ \left \{ \ln \left ( x \right ) -{\frac {1}{a}\sqrt {-{a}^{2}}\arctan \left ( {\frac {{a}^{2}y \left ( x \right ) }{x}{\frac {1}{\sqrt {-{a}^{ 2}}}}{\frac {1}{\sqrt {-{\frac {{a}^{2}{x}^{2}-{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}}}} \right ) }+{\frac {1}{2} \ln \left ( {\frac { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}{ {x}^{2}}} \right ) }+{\frac {1}{a}\ln \left ( {\frac {1}{x} \left ( \sqrt {{\frac {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}x+y \left ( x \right ) \right ) } \right ) }-{ \it \_C1}=0,\ln \left ( x \right ) +{\frac {1}{a}\sqrt {-{a}^{2}} \arctan \left ( {\frac {{a}^{2}y \left ( x \right ) }{x}{\frac {1}{\sqrt {-{a}^{2}}}}{\frac {1}{\sqrt {-{\frac {{a}^{2}{x}^{2}-{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}}}} \right ) }+{\frac {1}{2 }\ln \left ( {\frac { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2} }{{x}^{2}}} \right ) }-{\frac {1}{a}\ln \left ( {\frac {1}{x} \left ( \sqrt {{\frac {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}x+y \left ( x \right ) \right ) } \right ) }-{ \it \_C1}=0 \right \} \]