\[ 4 y(x)^2 (1-y(x)) \left (-f'(x)+f(x)^2-g'(x)-g(x)^2\right )-4 y(x) y'(x) (f(x) y(x)+g(x))+(1-y(x))^3 \left (\text {f0}(x)^2 y(x)^2-\text {f1}(x)^2\right )-2 y(x) (1-y(x)) y''(x)+(1-3 y(x)) y'(x)^2=0 \] ✗ Mathematica : cpu = 11.0732 (sec), leaf count = 0 , could not solve
DSolve[(1 - y[x])^3*(-f1[x]^2 + f0[x]^2*y[x]^2) + 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 , could not solve
dsolve(-2*y(x)*(1-y(x))*diff(diff(y(x),x),x)+(1-3*y(x))*diff(y(x),x)^2-4*y(x)*diff(y(x),x)*(f(x)*y(x)+g(x))+(1-y(x))^3*(f0(x)^2*y(x)^2-f1(x)^2)+4*y(x)^2*(1-y(x))*(f(x)^2-g(x)^2-diff(g(x),x)-diff(f(x),x))=0,y(x))