\[ \boxed { -2\,y \left ( x \right ) \left ( 1-y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( 1-3\,y \left ( x \right ) \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-4\,y \left ( x \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) \left ( f \left ( x \right ) y \left ( x \right ) +g \left ( x \right ) \right ) + \left ( 1-y \left ( x \right ) \right ) ^{3} \left ( \left ( {\it f0} \left ( x \right ) \right ) ^{2} \left ( y \left ( x \right ) \right ) ^{2}- \left ( {\it f1} \left ( x \right ) \right ) ^{2} \right ) +4\, \left ( y \left ( x \right ) \right ) ^{2} \left ( 1-y \left ( x \right ) \right ) \left ( \left ( f \left ( x \right ) \right ) ^{2}- \left ( g \left ( x \right ) \right ) ^{2}-{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) -{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 8.823120 (sec), leaf count = 110 \[ \text {DSolve}\left [4 y(x)^2 (1-y(x)) \left (-f'(x)+f(x)^2-g'(x)-g(x)^2\right )-4 y(x) y'(x) (f(x) y(x)+g(x))+(1-y(x))^3 \left (\text {f0}(x)^2 y(x)^2-\text {f1}(x)^2\right )-2 y(x) (1-y(x)) y''(x)+(1-3 y(x)) y'(x)^2=0,y(x),x\right ] \]
Maple: cpu = 11.169 (sec), leaf count = 0 \[ \text {could not solve} \]