\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-F \left ( x \right ) \left ( -{x}^{2}-2\,xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2} \right ) +{\frac {y \left ( x \right ) }{x}}=0} \]
Mathematica: cpu = 0.365546 (sec), leaf count = 104 \[ \left \{\left \{y(x)\to -\frac {x \left (-\exp \left (2 \sqrt {2} \left (\int _1^x K[1] (-F(K[1])) \, dK[1]+c_1\right )\right )+\sqrt {2} \exp \left (2 \sqrt {2} \left (\int _1^x K[1] (-F(K[1])) \, dK[1]+c_1\right )\right )-1-\sqrt {2}\right )}{\exp \left (2 \sqrt {2} \left (\int _1^x K[1] (-F(K[1])) \, dK[1]+c_1\right )\right )+1}\right \}\right \} \]
Maple: cpu = 0.031 (sec), leaf count = 29 \[ \left \{ y \left ( x \right ) ={\frac {x \left ( \sqrt {2}+2\,\tanh \left ( \left ( \int \!F \left ( x \right ) x\,{\rm d}x+{\it \_C1} \right ) \sqrt {2} \right ) \right ) \sqrt {2}}{2}} \right \} \]