\[ \boxed { 2\,y \left ( x \right ) \left ( -1+y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( 3\,y \left ( x \right ) -1 \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+4\,y \left ( x \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) \left ( f \left ( x \right ) y \left ( x \right ) +g \left ( x \right ) \right ) +4\, \left ( y \left ( x \right ) \right ) ^{2} \left ( -1+y \left ( x \right ) \right ) \left ( \left ( g \left ( x \right ) \right ) ^{2}- \left ( f \left ( x \right ) \right ) ^{2}-{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) -{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 1.564699 (sec), leaf count = 85 \[ \text {DSolve}\left [-4 (1-y(x)) y(x)^2 \left (-f'(x)-f(x)^2-g'(x)+g(x)^2\right )+4 y(x) y'(x) (f(x) y(x)+g(x))-2 (1-y(x)) y(x) y''(x)+(1-3 y(x)) y'(x)^2=0,y(x),x\right ] \]
Maple: cpu = 0.219 (sec), leaf count = 113 \[ \left \{ \sqrt {y \left ( x \right ) }-2\,{\frac {{\frac {\partial }{ \partial x}}{\it DESol} \left ( \left \{ -1/4\,{{\rm e}^{-2\,\int \!f \left ( x \right ) \,{\rm d}x+2\,\int \!g \left ( x \right ) \,{\rm d}x}} {{\it \_C1}}^{2}{\it \_Y} \left ( x \right ) -2\,g \left ( x \right ) { \frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) +{\frac {{\rm d}^{2 }}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) \right \} , \left \{ { \it \_Y} \left ( x \right ) \right \} \right ) }{{\it DESol} \left ( \left \{ -1/4\,{{\rm e}^{-2\,\int \!f \left ( x \right ) \,{\rm d}x+2\, \int \!g \left ( x \right ) \,{\rm d}x}}{{\it \_C1}}^{2}{\it \_Y} \left ( x \right ) -2\,g \left ( x \right ) {\frac {\rm d}{{\rm d}x}}{ \it \_Y} \left ( x \right ) +{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) {{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x+ \int \!g \left ( x \right ) \,{\rm d}x}}{\it \_C1}}}=0 \right \} \]