\[ -f(x) ((y(x)-1) y(x) (y(x)-x))^{3/2}+2 (1-y(x)) \left (x^2-2 x y(x)+y(x)\right ) y(x) y'(x)-2 (1-x) x (1-y(x)) (x-y(x)) y(x) y''(x)+(1-x) x \left (3 y(x)^2-2 x y(x)-2 y(x)+x\right ) y'(x)^2-(1-y(x))^2 y(x)^2=0 \] ✗ Mathematica : cpu = 8.57217 (sec), leaf count = 0
DSolve[-((1 - y[x])^2*y[x]^2) - f[x]*((-1 + y[x])*y[x]*(-x + y[x]))^(3/2) + 2*(1 - y[x])*y[x]*(x^2 + y[x] - 2*x*y[x])*Derivative[1][y][x] + (1 - x)*x*(x - 2*y[x] - 2*x*y[x] + 3*y[x]^2)*Derivative[1][y][x]^2 - 2*(1 - x)*x*(1 - y[x])*(x - y[x])*y[x]*Derivative[2][y][x] == 0,y[x],x]
, could not solve
DSolve[-((1 - y[x])^2*y[x]^2) - f[x]*((-1 + y[x])*y[x]*(-x + y[x]))^(3/2) + 2*(1 - y[x])*y[x]*(x^2 + y[x] - 2*x*y[x])*Derivative[1][y][x] + (1 - x)*x*(x - 2*y[x] - 2*x*y[x] + 3*y[x]^2)*Derivative[1][y][x]^2 - 2*(1 - x)*x*(1 - y[x])*(x - y[x])*y[x]*Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 2.646 (sec), leaf count = 726
dsolve(-2*x*y(x)*(1-x)*(1-y(x))*(x-y(x))*diff(diff(y(x),x),x)+x*(1-x)*(x-2*x*y(x)-2*y(x)+3*y(x)^2)*diff(y(x),x)^2+2*y(x)*(1-y(x))*(x^2+y(x)-2*x*y(x))*diff(y(x),x)-y(x)^2*(1-y(x))^2-f*(y(x)*(-1+y(x))*(y(x)-x))^(3/2)=0,y(x))
\[\text {Expression too large to display}\]