\[ -4 (1-y(x)) y(x)^2 \left (-f'(x)-f(x)^2-g'(x)+g(x)^2\right )+4 y(x) y'(x) (f(x) y(x)+g(x))-2 (1-y(x)) y(x) y''(x)+(1-3 y(x)) y'(x)^2=0 \] ✗ Mathematica : cpu = 1.2117 (sec), leaf count = 0
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]
, 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
dsolve(2*y(x)*(-1+y(x))*diff(diff(y(x),x),x)-(3*y(x)-1)*diff(y(x),x)^2+4*y(x)*diff(y(x),x)*(f(x)*y(x)+g(x))+4*y(x)^2*(-1+y(x))*(g(x)^2-f(x)^2-diff(g(x),x)-diff(f(x),x))=0,y(x))
, result contains DESol or ODESolStruc
\[\sqrt {y \left (x \right )}-\frac {2 \left (\frac {\partial }{\partial x}\mathit {DESol}\left (\left \{-\frac {{\mathrm e}^{2 \left (\int g \left (x \right )d x \right )-2 \left (\int f \left (x \right )d x \right )} c_{1}^{2} \textit {\_Y} \left (x \right )}{4}-2 g \left (x \right ) \left (\frac {d}{d x}\textit {\_Y} \left (x \right )\right )+\frac {d^{2}}{d x^{2}}\textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) {\mathrm e}^{\int \left (f \left (x \right )-g \left (x \right )\right )d x}}{\mathit {DESol}\left (\left \{-\frac {{\mathrm e}^{2 \left (\int g \left (x \right )d x \right )-2 \left (\int f \left (x \right )d x \right )} c_{1}^{2} \textit {\_Y} \left (x \right )}{4}-2 g \left (x \right ) \left (\frac {d}{d x}\textit {\_Y} \left (x \right )\right )+\frac {d^{2}}{d x^{2}}\textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) c_{1}} = 0\]