\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-{\frac {27\,xy \left ( x \right ) }{16\, \left ( {x}^{3}-1 \right ) ^{2}}}=0} \]
Mathematica: cpu = 276.981672 (sec), leaf count = 257 \[ \left \{\left \{y(x)\to \frac {\sqrt {2} c_2 (1-x)^{3/4} \sqrt [4]{x^2+x+1} \int _1^x \frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}} \, dK[1]}{\sqrt [4]{\sqrt {3} x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+\sqrt {3}}}+\frac {\sqrt {2} c_1 (1-x)^{3/4} \sqrt [4]{x^2+x+1}}{\sqrt [4]{\sqrt {3} x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+\sqrt {3}}}\right \}\right \} \]
Maple: cpu = 0.109 (sec), leaf count = 53 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sqrt {x}\sqrt [4]{{x}^{3}-1} {\it LegendreP} \left ( -{\frac {1}{6}},{\frac {1}{3}},\sqrt {-{x}^{3}+ 1} \right ) +{\it \_C2}\,\sqrt {x}\sqrt [4]{{x}^{3}-1}{\it LegendreQ} \left ( -{\frac {1}{6}},{\frac {1}{3}},\sqrt {-{x}^{3}+1} \right ) \right \} \]