\[ \boxed { \left ( -{a}^{2}+1 \right ) \left ( y \left ( x \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-2\,{a}^{2}xy \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2}{x}^{2}=0} \]
Mathematica: cpu = 0.333542 (sec), leaf count = 212 \[ \left \{\left \{y(x)\to -\frac {\sqrt {a^6 \left (-x^2\right )+3 a^4 x^2+2 a^2 x e^{a^2 c_1-c_1}-2 x e^{a^2 c_1-c_1}+e^{2 a^2 c_1-2 c_1}-3 a^2 x^2+x^2}}{\sqrt {a^6-3 a^4+3 a^2-1}}\right \},\left \{y(x)\to \frac {\sqrt {a^6 \left (-x^2\right )+3 a^4 x^2+2 a^2 x e^{a^2 c_1-c_1}-2 x e^{a^2 c_1-c_1}+e^{2 a^2 c_1-2 c_1}-3 a^2 x^2+x^2}}{\sqrt {a^6-3 a^4+3 a^2-1}}\right \}\right \} \]
Maple: cpu = 0.702 (sec), leaf count = 201 \[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) +\int ^{{\it \_Z}}\!{\frac {{\it \_a}}{{a}^{2}{{\it \_a}}^{4} -{{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2}} \left ( -{{\it \_a}}^{2}{a}^{2}+{{\it \_a}}^{2}-{a}^{2}+\sqrt {{{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2}} \right ) }{d{\it \_a}}+{\it \_C1} \right ) x,y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) -\int ^{{\it \_Z}}\!{\frac {{\it \_a}}{{a}^{2}{{\it \_a}}^{4} -{{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2}} \left ( {{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2}+\sqrt {{{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2}} \right ) }{d{\it \_a}}+{\it \_C1} \right ) x,y \left ( x \right ) ={ax{\frac {1}{\sqrt {-{a}^{2}+1}}} },y \left ( x \right ) =-{ax{\frac {1}{\sqrt {-{a}^{2}+1}}}} \right \} \]