Internal problem ID [8985]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1406.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {27 x y}{16 \left (x^{3}-1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 53
dsolve(diff(diff(y(x),x),x) = -27/16*x/(x^3-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (x^{3}-1\right )^{\frac {1}{4}} \LegendreP \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+c_{2} \sqrt {x}\, \left (x^{3}-1\right )^{\frac {1}{4}} \LegendreQ \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \]
✓ Solution by Mathematica
Time used: 31.541 (sec). Leaf size: 1150
DSolve[y''[x] == (-27*x*y[x])/(16*(-1 + x^3)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{-\frac {1}{2} \tanh ^{-1}\left (\frac {-2 x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+2}{2 \sqrt {3}}\right )} (x-1) \left (\left (3974 i-4610 \sqrt {3}\right ) x+\left (-1987 i+2305 \sqrt {3}\right ) \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-2 \sqrt {2 \left (-9867779-707709 i \sqrt {3}\right )}\right )^{\frac {1}{32} \left (11+5 i \sqrt {3}\right )} \left (-2 \left (-i+\sqrt {3}\right ) x-i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+\sqrt {2 x+i \sqrt {3}+1} \sqrt {6 x-3 i \sqrt {3}+3}+4 i\right )^{\frac {5}{32} \left (1-i \sqrt {3}\right )} \left (-2 \left (9 i+5 \sqrt {3}\right ) x+9 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+5 \sqrt {2 x+i \sqrt {3}+1} \sqrt {6 x-3 i \sqrt {3}+3}+4 \sqrt {3}-24 i\right )^{\frac {1}{32} \left (1+i \sqrt {3}\right )} \left (\left (256985009468147850 i-163072339831481354 \sqrt {3}\right ) x-128492504734073925 i \sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+81536169915740677 \sqrt {2 x+i \sqrt {3}+1} \sqrt {6 x-3 i \sqrt {3}+3}-210028674649814602 \sqrt {3}-116116005013148106 i\right )^{\frac {1}{32} \left (15-i \sqrt {3}\right )} \left (c_1+c_2 \int _1^x\frac {2 e^{\tanh ^{-1}\left (\frac {-2 K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+2}{2 \sqrt {3}}\right )} \left (-2 K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-1\right ) \left (\left (3974 i-4610 \sqrt {3}\right ) K[1]+\left (-1987 i+2305 \sqrt {3}\right ) \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}-2 \sqrt {2 \left (-9867779-707709 i \sqrt {3}\right )}\right )^{\frac {1}{16} \left (-11-5 i \sqrt {3}\right )} \left (-2 \left (-i+\sqrt {3}\right ) K[1]-i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+4 i\right )^{\frac {5}{16} i \left (i+\sqrt {3}\right )} \left (-2 \left (9 i+5 \sqrt {3}\right ) K[1]+9 i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+5 \sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+4 \sqrt {3}-24 i\right )^{-\frac {1}{16} i \left (-i+\sqrt {3}\right )} \left (\left (256985009468147850 i-163072339831481354 \sqrt {3}\right ) K[1]-128492504734073925 i \sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+81536169915740677 \sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}-210028674649814602 \sqrt {3}-116116005013148106 i\right )^{\frac {1}{16} i \left (15 i+\sqrt {3}\right )}}{K[1]-1}dK[1]\right )}{\sqrt {2} \sqrt {(x-1) \left (-2 x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}-1\right )}} \\ \end{align*}