ODE
\[ 2 (1-y(x)) y(x) y''(x)=f(x) (1-y(x)) y(x) y'(x)+(1-2 y(x)) y'(x)^2 \] ODE Classification
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.0418 (sec), leaf count = 87
\[\left \{\left \{y(x)\to \frac {1}{4} \exp \left (-i \left (\int _1^x c_1 \left (-e^{-\int _1^{K[3]} -\frac {1}{2} f(K[1]) \, dK[1]}\right ) \, dK[3]+c_2\right )\right ) \left (1+\exp \left (i \left (\int _1^x c_1 \left (-e^{-\int _1^{K[3]} -\frac {1}{2} f(K[1]) \, dK[1]}\right ) \, dK[3]+c_2\right )\right )\right ){}^2\right \}\right \}\]
Maple ✓
cpu = 0.053 (sec), leaf count = 37
\[ \left \{ \ln \left ( -{\frac {1}{2}}+y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-y \left ( x \right ) } \right ) -{\it \_C1}\,\int \!{{\rm e}^{-\int \!-{\frac {f \left ( x \right ) }{2}}\,{\rm d}x}}\,{\rm d}x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[2*(1 - y[x])*y[x]*y''[x] == f[x]*(1 - y[x])*y[x]*y'[x] + (1 - 2*y[x])*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> (1 + E^(I*(C[2] + Integrate[-(C[1]/E^Integrate[-f[K[1]]/2, {K[1], 1, K[3]}]), {K[3], 1, x}])))^2/(4*E^(I*(C[2] + Integrate[-(C[1]/E^Integrate[-f[K[1]]/2, {K[1], 1, K[3]}]), {K[3], 1, x}])))}}
Maple raw input
dsolve(2*y(x)*(1-y(x))*diff(diff(y(x),x),x) = (1-2*y(x))*diff(y(x),x)^2+y(x)*(1-y(x))*diff(y(x),x)*f(x), y(x),'implicit')
Maple raw output
ln(-1/2+y(x)+(y(x)^2-y(x))^(1/2))-_C1*Int(exp(-Int(-1/2*f(x),x)),x)-_C2 = 0