\[ \boxed { \left ( F_{{1,1}} \right ) \left ( x \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+ \left ( \left ( \left ( F_{{2,1}} \right ) \left ( x \right ) + \left ( F_{{1,2}} \right ) \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +y \left ( x \right ) \left ( \left ( F_{{1,0}} \right ) \left ( x \right ) + \left ( F_{{0,1}} \right ) \left ( x \right ) \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( F_{{2,2}} \right ) \left ( x \right ) \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) ^{2}+y \left ( x \right ) \left ( \left ( F_{{2,0}} \right ) \left ( x \right ) + \left ( F_{{0,2}} \right ) \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( F_{{0,0}} \right ) \left ( x \right ) \left ( y \left ( x \right ) \right ) ^{2}=0} \]
Mathematica: cpu = 230.453764 (sec), leaf count = 90 \[ \text {DSolve}\left [y(x) (x F(0,2)+x F(2,0)) y''(x)+x F(2,2) y''(x)^2+x F(1,1) y''(x)+y'(x) \left ((x F(1,2)+x F(2,1)) y''(x)+y(x) (x F(0,1)+x F(1,0))\right )+x F(0,0) y(x)^2=0,y(x),x\right ] \]
Maple: cpu = 1.731 (sec), leaf count = 191 \[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =- \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}-{ \frac { \left ( \left ( F_{{2,1}} \right ) \left ( {\it \_a} \right ) + \left ( F_{{1,2}} \right ) \left ( {\it \_a} \right ) \right ) {\it \_b} \left ( {\it \_a} \right ) }{2\, \left ( F_{{2,2}} \right ) \left ( {\it \_a} \right ) }}-{\frac {1}{2\, \left ( F_{{2,2}} \right ) \left ( {\it \_a} \right ) } \left ( \left ( F_{{2,0}} \right ) \left ( {\it \_a} \right ) + \left ( F_{{0,2}} \right ) \left ( {\it \_a} \right ) -\sqrt { -4\, \left ( F_{{2,2}} \right ) \left ( {\it \_a} \right ) \left ( F_{{0,0 }} \right ) \left ( {\it \_a} \right ) + \left ( \left ( F_{{2,0}} \right ) \left ( {\it \_a} \right ) \right ) ^{2}+2\, \left ( F_{{2,0}} \right ) \left ( {\it \_a} \right ) \left ( F_{{0,2}} \right ) \left ( { \it \_a} \right ) + \left ( \left ( F_{{0,2}} \right ) \left ( {\it \_a} \right ) \right ) ^{2}+2\, \left ( F_{{2,0}} \right ) \left ( {\it \_a} \right ) \left ( F_{{2,1}} \right ) \left ( {\it \_a} \right ) {\it \_b} \left ( {\it \_a} \right ) +2\,{\it \_b} \left ( {\it \_a} \right ) \left ( F_{{2,1}} \right ) \left ( {\it \_a} \right ) \left ( F_{{0,2}} \right ) \left ( {\it \_a} \right ) +2\, \left ( F_{{2,0}} \right ) \left ( {\it \_a} \right ) \left ( F_{{1,2}} \right ) \left ( {\it \_a} \right ) {\it \_b} \left ( {\it \_a} \right ) +2\, \left ( F_{{0,2}} \right ) \left ( {\it \_a} \right ) \left ( F_{{1,2}} \right ) \left ( { \it \_a} \right ) {\it \_b} \left ( {\it \_a} \right ) -4\,{\it \_b} \left ( {\it \_a} \right ) \left ( F_{{2,2}} \right ) \left ( {\it \_a} \right ) \left ( F_{{1,0}} \right ) \left ( {\it \_a} \right ) -4\,{\it \_b} \left ( {\it \_a} \right ) \left ( F_{{0,1}} \right ) \left ( {\it \_a} \right ) \left ( F_{{2,2}} \right ) \left ( {\it \_a} \right ) -4\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \left ( F_{{2, 2}} \right ) \left ( {\it \_a} \right ) \left ( F_{{1,1}} \right ) \left ( {\it \_a} \right ) + \left ( \left ( F_{{2,1}} \right ) \left ( { \it \_a} \right ) \right ) ^{2} \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}+2\, \left ( F_{{2,1}} \right ) \left ( {\it \_a} \right ) \left ( F_{{1,2}} \right ) \left ( {\it \_a} \right ) \left ( { \it \_b} \left ( {\it \_a} \right ) \right ) ^{2}+ \left ( \left ( F_{{1, 2}} \right ) \left ( {\it \_a} \right ) \right ) ^{2} \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}} \right ) } \right \} , \left \{ {\it \_a}=x,{\it \_b} \left ( {\it \_a} \right ) ={\frac {{\frac {\rm d} {{\rm d}x}}y \left ( x \right ) }{y \left ( x \right ) }} \right \} , \left \{ x={\it \_a},y \left ( x \right ) ={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right \} ] \right ) \right \} \]