\[ -4 (1-y(x)) y(x)^2 \left (-f'(x)-f(x)^2-g'(x)+g(x)^2\right )+4 y(x) y'(x) (f(x) y(x)+g(x))+(1-3 y(x)) y'(x)^2-2 (1-y(x)) y(x) y''(x)=0 \] ✗ Mathematica : cpu = 1.14848 (sec), leaf count = 0 , could not solve
DSolve[-4*(1 - y[x])*y[x]^2*(-f[x]^2 + g[x]^2 - Derivative[1][f][x] - Derivative[1][g][x]) + 4*y[x]*(g[x] + f[x]*y[x])*Derivative[1][y][x] + (1 - 3*y[x])*Derivative[1][y][x]^2 - 2*(1 - y[x])*y[x]*Derivative[2][y][x] == 0, y[x], x]
✗ Maple : cpu = 0. (sec), leaf count = 0 , result contains DESol
\[ \left \{ \sqrt {y \left ( x \right ) }-2\,{\frac {{\frac {\partial }{\partial x}}{\it DESol} \left ( \left \{ -1/4\,{{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x-2\,\int \!f \left ( x \right ) \,{\rm d}x}}{{\it \_C1}}^{2}{\it \_Y} \left ( x \right ) -2\,g \left ( x \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) +{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) }{{\it DESol} \left ( \left \{ -1/4\,{{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x-2\,\int \!f \left ( x \right ) \,{\rm d}x}}{{\it \_C1}}^{2}{\it \_Y} \left ( x \right ) -2\,g \left ( x \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) +{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) {{\rm e}^{\int \!g \left ( x \right ) \,{\rm d}x-\int \!f \left ( x \right ) \,{\rm d}x}}{\it \_C1}}}=0 \right \} \]