ODE
\[ 3 (1-y(x)) y(x) y''(x)=2 (1-2 y(x)) y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.335769 (sec), leaf count = 53
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {3 (1-\text {$\#$1})^{2/3} \text {$\#$1} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\text {$\#$1}\right )}{(-(\text {$\#$1}-1) \text {$\#$1})^{2/3} c_1}\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.191 (sec), leaf count = 41
\[ \left \{ 3\,{\frac { \left ( -{\it signum} \left ( y \left ( x \right ) -1 \right ) \right ) ^{2/3}\sqrt [3]{y \left ( x \right ) }{\mbox {$_2$F$_1$}(1/3,2/3;\,4/3;\,y \left ( x \right ) )}}{ \left ( {\it signum} \left ( y \left ( x \right ) -1 \right ) \right ) ^{2/3}}}-{\it \_C1}\,x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[3*(1 - y[x])*y[x]*y''[x] == 2*(1 - 2*y[x])*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(3*Hypergeometric2F1[1/3, 2/3, 4/3, #1]*(1 - #1)^(2/3)*#1)/(C[1]*(-((-1 + #1)*#1))^(2/3)) & ][x + C[2]]}}
Maple raw input
dsolve(3*y(x)*(1-y(x))*diff(diff(y(x),x),x) = 2*(1-2*y(x))*diff(y(x),x)^2, y(x),'implicit')
Maple raw output
3/signum(y(x)-1)^(2/3)*(-signum(y(x)-1))^(2/3)*y(x)^(1/3)*hypergeom([1/3, 2/3],[4/3],y(x))-_C1*x-_C2 = 0