\[ \boxed { 2\,{x}^{2}y \left ( x \right ) \left ( 1-x \right ) ^{2} \left ( 1-y \left ( x \right ) \right ) \left ( x-y \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -{x}^{2} \left ( 1-x \right ) ^{2} \left ( x-2\,xy \left ( x \right ) -2\,y \left ( x \right ) +3\, \left ( y \left ( x \right ) \right ) ^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-2\,xy \left ( x \right ) \left ( 1-x \right ) \left ( 1-y \left ( x \right ) \right ) \left ( {x}^{2}+y \left ( x \right ) -2\,xy \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +bx \left ( 1-y \left ( x \right ) \right ) ^{2} \left ( x-y \left ( x \right ) \right ) ^{2}-c \left ( 1-x \right ) \left ( y \left ( x \right ) \right ) ^{2} \left ( x-y \left ( x \right ) \right ) ^{2}-dx \left ( y \left ( x \right ) \right ) ^{2} \left ( 1-x \right ) \left ( 1-y \left ( x \right ) \right ) ^{2}+a \left ( y \left ( x \right ) \right ) ^{2} \left ( x-y \left ( x \right ) \right ) ^{2} \left ( 1-y \left ( x \right ) \right ) ^{2}=0} \]
Mathematica: cpu = 33.961813 (sec), leaf count = 185 \[ \text {DSolve}\left [a (1-y(x))^2 (x-y(x))^2 y(x)^2+b x (1-y(x))^2 (x-y(x))^2-c (1-x) (x-y(x))^2 y(x)^2-d (1-x) x (1-y(x))^2 y(x)^2+2 (1-x)^2 x^2 (1-y(x)) (x-y(x)) y(x) y''(x)-(1-x)^2 x^2 \left (3 y(x)^2-2 x y(x)-2 y(x)+x\right ) y'(x)^2-2 (1-x) x (1-y(x)) y(x) \left (x^2-2 x y(x)+y(x)\right ) y'(x)=0,y(x),x\right ] \]
Maple: cpu = 136.267 (sec), leaf count = 0 \[ \text {could not solve} \]