#### 2.499   ODE No. 499

$a^2 \left (-x^2\right )-2 a^2 x y(x) y'(x)+\left (1-a^2\right ) y(x)^2 y'(x)^2+y(x)^2=0$ Mathematica : cpu = 0.363784 (sec), leaf count = 212

$\left \{\left \{y(x)\to -\frac {\sqrt {a^6 \left (-x^2\right )+3 a^4 x^2-3 a^2 x^2+2 a^2 x e^{a^2 c_1-c_1}-2 x e^{a^2 c_1-c_1}+e^{2 a^2 c_1-2 c_1}+x^2}}{\sqrt {a^6-3 a^4+3 a^2-1}}\right \},\left \{y(x)\to \frac {\sqrt {a^6 \left (-x^2\right )+3 a^4 x^2-3 a^2 x^2+2 a^2 x e^{a^2 c_1-c_1}-2 x e^{a^2 c_1-c_1}+e^{2 a^2 c_1-2 c_1}+x^2}}{\sqrt {a^6-3 a^4+3 a^2-1}}\right \}\right \}$ Maple : cpu = 0.238 (sec), leaf count = 189

$\left \{ y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) +\int ^{{\it \_Z}}\!{\frac {{\it \_a}}{ \left ( {{\it \_a}}^{2}+1 \right ) \left ( {{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2} \right ) } \left ( -{{\it \_a}}^{2}{a}^{2}+{{\it \_a}}^{2}-{a}^{2}+\sqrt {{{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2}} \right ) }{d{\it \_a}}+{\it \_C1} \right ) x,y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) -\int ^{{\it \_Z}}\!{\frac {{\it \_a}}{ \left ( {{\it \_a}}^{2}+1 \right ) \left ( {{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2} \right ) } \left ( {{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2}+\sqrt {{{\it \_a}}^{2}{a}^{2}-{{\it \_a}}^{2}+{a}^{2}} \right ) }{d{\it \_a}}+{\it \_C1} \right ) x,y \left ( x \right ) ={ax{\frac {1}{\sqrt {-{a}^{2}+1}}}},y \left ( x \right ) =-{ax{\frac {1}{\sqrt {-{a}^{2}+1}}}} \right \}$