\[ y'(x)=\frac {x F\left (\frac {x^2 y(x)+\frac {1}{4}}{x^2}\right )+\frac {1}{2}}{x^3} \] ✓ Mathematica : cpu = 0.253463 (sec), leaf count = 144
DSolve[Derivative[1][y][x] == (1/2 + x*F[(1/4 + x^2*y[x])/x^2])/x^3,y[x],x]
\[\text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {K[2] x^2+\frac {1}{4}}{x^2}\right ) \int _1^x-\frac {F'\left (\frac {K[2] K[1]^2+\frac {1}{4}}{K[1]^2}\right )}{2 F\left (\frac {K[2] K[1]^2+\frac {1}{4}}{K[1]^2}\right )^2 K[1]^3}dK[1]+1}{F\left (\frac {K[2] x^2+\frac {1}{4}}{x^2}\right )}dK[2]+\int _1^x\left (\frac {1}{K[1]^2}+\frac {1}{2 K[1]^3 F\left (\frac {y(x) K[1]^2+\frac {1}{4}}{K[1]^2}\right )}\right )dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.312 (sec), leaf count = 32
dsolve(diff(y(x),x) = 1/2*(1+2*F(1/4*(4*x^2*y(x)+1)/x^2)*x)/x^3,y(x))
\[y \left (x \right ) = \frac {4 \RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} \right ) x +x c_{1}+1\right ) x^{2}-1}{4 x^{2}}\]