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: 0.014 (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: 51.058 (sec). Leaf size: 180
DSolve[y''[x] == (-27*x*y[x])/(16*(-1 + x^3)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {2} (1-x)^{3/4} \sqrt [4]{x^2+x+1} \left (c_2 \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]+c_1\right )}{\sqrt [4]{\sqrt {3} x+\sqrt {2 x-i \sqrt {3}+1} \sqrt {2 x+i \sqrt {3}+1}+\sqrt {3}}} \\ \end{align*}