\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =F \left ( {\frac {y \left ( x \right ) }{x+a}} \right ) =0} \]
Mathematica: cpu = 12.111538 (sec), leaf count = 240 \[ \text {Solve}\left [\int _1^{y(x)} \left (\frac {1}{-a F\left (\frac {K[2]}{a+x}\right )-x F\left (\frac {K[2]}{a+x}\right )+K[2]}-\int _1^x \left (\frac {F'\left (\frac {K[2]}{K[1]+a}\right )}{(K[1]+a) \left (a F\left (\frac {K[2]}{K[1]+a}\right )+K[1] F\left (\frac {K[2]}{K[1]+a}\right )-K[2]\right )}-\frac {F\left (\frac {K[2]}{K[1]+a}\right ) \left (\frac {a F'\left (\frac {K[2]}{K[1]+a}\right )}{K[1]+a}+\frac {K[1] F'\left (\frac {K[2]}{K[1]+a}\right )}{K[1]+a}-1\right )}{\left (a F\left (\frac {K[2]}{K[1]+a}\right )+K[1] F\left (\frac {K[2]}{K[1]+a}\right )-K[2]\right )^2}\right ) \, dK[1]\right ) \, dK[2]+\int _1^x \frac {F\left (\frac {y(x)}{K[1]+a}\right )}{a F\left (\frac {y(x)}{K[1]+a}\right )+K[1] F\left (\frac {y(x)}{K[1]+a}\right )-y(x)} \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.031 (sec), leaf count = 28 \[ \left \{ y \left ( x \right ) =-{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( F \left ( -{\it \_a} \right ) +{\it \_a} \right ) ^{-1}{d{\it \_a }}+\ln \left ( x+a \right ) +{\it \_C1} \right ) \left ( x+a \right ) \right \} \]