\[ y(x) (x F(0,2)+x F(2,0)) y''(x)+x F(2,2) y''(x)^2+x F(1,1) y''(x)+y'(x) \left ((x F(1,2)+x F(2,1)) y''(x)+y(x) (x F(0,1)+x F(1,0))\right )+x F(0,0) y(x)^2=0 \] ✗ Mathematica : cpu = 71.5522 (sec), leaf count = 0
DSolve[x*F[0, 0]*y[x]^2 + x*F[1, 1]*Derivative[2][y][x] + (x*F[0, 2] + x*F[2, 0])*y[x]*Derivative[2][y][x] + x*F[2, 2]*Derivative[2][y][x]^2 + Derivative[1][y][x]*((x*F[0, 1] + x*F[1, 0])*y[x] + (x*F[1, 2] + x*F[2, 1])*Derivative[2][y][x]) == 0,y[x],x]
, could not solve
DSolve[x*F[0, 0]*y[x]^2 + x*F[1, 1]*Derivative[2][y][x] + (x*F[0, 2] + x*F[2, 0])*y[x]*Derivative[2][y][x] + x*F[2, 2]*Derivative[2][y][x]^2 + Derivative[1][y][x]*((x*F[0, 1] + x*F[1, 0])*y[x] + (x*F[1, 2] + x*F[2, 1])*Derivative[2][y][x]) == 0, y[x], x]
✗ Maple : cpu = 0. (sec), leaf count = 0
dsolve(F[1,1](x)*diff(y(x),x)^2+((F[2,1](x)+F[1,2](x))*diff(diff(y(x),x),x)+y(x)*(F[1,0](x)+F[0,1](x)))*diff(y(x),x)+F[2,2](x)*diff(diff(y(x),x),x)^2+y(x)*(F[2,0](x)+F[0,2](x))*diff(diff(y(x),x),x)+F[0,0](x)*y(x)^2=0,y(x))
, result contains DESol or ODESolStruc
\[y \left (x \right ) = \left ({\mathrm e}^{\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}}\right )\boldsymbol {\mathrm {where}}\left [\left \{\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )=-\textit {\_}b\left (\textit {\_a} \right )^{2}-\frac {\left (F_{2,1}\left (\textit {\_a} \right )+F_{1,2}\left (\textit {\_a} \right )\right ) \textit {\_}b\left (\textit {\_a} \right )}{2 F_{2,2}\left (\textit {\_a} \right )}-\frac {F_{2,0}\left (\textit {\_a} \right )+F_{0,2}\left (\textit {\_a} \right )-\sqrt {F_{2,0}\left (\textit {\_a} \right )^{2}+2 F_{2,0}\left (\textit {\_a} \right ) F_{0,2}\left (\textit {\_a} \right )+2 F_{2,1}\left (\textit {\_a} \right ) F_{2,0}\left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )+2 F_{1,2}\left (\textit {\_a} \right ) F_{2,0}\left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )+F_{0,2}\left (\textit {\_a} \right )^{2}+2 F_{2,1}\left (\textit {\_a} \right ) F_{0,2}\left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )+2 F_{1,2}\left (\textit {\_a} \right ) F_{0,2}\left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )-4 F_{0,0}\left (\textit {\_a} \right ) F_{2,2}\left (\textit {\_a} \right )+F_{2,1}\left (\textit {\_a} \right )^{2} \textit {\_}b\left (\textit {\_a} \right )^{2}+2 F_{2,1}\left (\textit {\_a} \right ) F_{1,2}\left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )^{2}+F_{1,2}\left (\textit {\_a} \right )^{2} \textit {\_}b\left (\textit {\_a} \right )^{2}-4 F_{1,0}\left (\textit {\_a} \right ) F_{2,2}\left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )-4 F_{0,1}\left (\textit {\_a} \right ) F_{2,2}\left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )-4 F_{1,1}\left (\textit {\_a} \right ) F_{2,2}\left (\textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )^{2}}}{2 F_{2,2}\left (\textit {\_a} \right )}\right \}, \left \{\textit {\_a} =x , \textit {\_}b\left (\textit {\_a} \right )=\frac {\frac {d}{d x}y \left (x \right )}{y \left (x \right )}\right \}, \left \{x =\textit {\_a} , y \left (x \right )={\mathrm e}^{\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}}\right \}\right ]\]