\[ y'(x)=\frac {(a-1) (a+1) x}{a^2 F\left (-\frac {1}{2} a^2 x^2+\frac {x^2}{2}+\frac {y(x)^2}{2}\right )-F\left (-\frac {1}{2} a^2 x^2+\frac {x^2}{2}+\frac {y(x)^2}{2}\right )+y(x)} \] ✓ Mathematica : cpu = 84.9394 (sec), leaf count = 144
\[\text {Solve}\left [c_1=\int _1^{y(x)} \left (-\int _1^x \frac {K[1] K[2] F'\left (\frac {1}{2} \left (K[2]^2-\left (a^2-1\right ) K[1]^2\right )\right )}{F\left (\frac {1}{2} \left (K[2]^2-\left (a^2-1\right ) K[1]^2\right )\right )^2} \, dK[1]+\frac {K[2]}{(a-1) (a+1) F\left (\frac {1}{2} \left (K[2]^2-a^2 x^2+x^2\right )\right )}+1\right ) \, dK[2]+\int _1^x -\frac {K[1]}{F\left (\frac {1}{2} \left (y(x)^2-\left (a^2-1\right ) K[1]^2\right )\right )} \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 3.014 (sec), leaf count = 60
\[ \left \{ {\frac {y \left ( x \right ) }{ \left ( a-1 \right ) \left ( a+1 \right ) }}+{\frac {1}{2\,{a}^{4}-4\,{a}^{2}+2}\int ^{-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}\! \left ( F \left ( {\frac {{\it \_a}}{2}} \right ) \right ) ^{-1}{d{\it \_a}}}-{\it \_C1}=0 \right \} \]