2.1406   ODE No. 1406

\[ y''(x)=-\frac {27 x y(x)}{16 \left (x^3-1\right )^2} \]

✓ Mathematica : cpu = 31.9758 (sec), leaf count = 707

DSolve[Derivative[2][y][x] == (-27*x*y[x])/(16*(-1 + x^3)^2),y[x],x]
 

\[\left \{\left \{y(x)\to \frac {i c_1 \sqrt [4]{2 x-i \sqrt {3}+1} \sqrt [4]{2 x+i \sqrt {3}+1} \left (\sqrt [6]{-1} \sqrt {2 x-i \sqrt {3}+1}-\sqrt {2 x+i \sqrt {3}+1}\right )^{3/4} \sqrt [4]{\sqrt [6]{-1} \sqrt {2 x-i \sqrt {3}+1}+\sqrt {2 x+i \sqrt {3}+1}}}{\sqrt {3}}-\frac {i c_2 \sqrt [4]{2 x-i \sqrt {3}+1} \sqrt [4]{2 x+i \sqrt {3}+1} \left (\sqrt [6]{-1} \sqrt {2 x-i \sqrt {3}+1}+\sqrt {2 x+i \sqrt {3}+1}\right )^{3/4} \left (\sqrt [6]{-1} \sqrt {2 x-i \sqrt {3}+1} \sqrt [4]{\sqrt [6]{-1} \sqrt {2 x-i \sqrt {3}+1}-\sqrt {2 x+i \sqrt {3}+1}}-\sqrt {2 x+i \sqrt {3}+1} \sqrt [4]{\sqrt [6]{-1} \sqrt {2 x-i \sqrt {3}+1}-\sqrt {2 x+i \sqrt {3}+1}}\right ) \left (\sqrt {2 x-i \sqrt {3}+1} +(-1)^{5/6} \sqrt {2 x-i \sqrt {3}+1} +\sqrt [6]{-1} \sqrt {2 x-i \sqrt {3}+1} +2 \sqrt [3]{-1} \sqrt {2 x+i \sqrt {3}+1} \right )}{2 \sqrt {2} \left (\sqrt [3]{-1}-1\right )^2 \left ((-1)^{5/6}-i\right ) \left (2 x +-2 +i \right )}\right \}\right \}\]

✓ Maple : cpu = 0.167 (sec), leaf count = 44

dsolve(diff(diff(y(x),x),x) = -27/16*x/(x^3-1)^2*y(x),y(x))
 

\[y \left (x \right ) = \sqrt {x}\, \left (x^{3}-1\right )^{{1}/{4}} \left (\operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) c_{2} +\operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) c_{1} \right )\]