Internal problem ID [7575]
Book: Collection of Kovacic problems
Section: section 3. Problems from Kovacic related papers
Problem number: Kovacic 1985 paper. page 25. section 5.2. Example 2.
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 {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 31
dsolve(diff(y(x),x$2)= -(5*x^2+27)/(36*(x^2-1)^2)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x^{2}-1}\, \LegendreP \left (-\frac {1}{6}, \frac {1}{3}, x\right )+c_{2} \sqrt {x^{2}-1}\, \LegendreQ \left (-\frac {1}{6}, \frac {1}{3}, x\right ) \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 38
DSolve[y''[x]== -(5*x^2+27)/(36*(x^2-1)^2)*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {x^2-1} \left (c_1 P_{-\frac {1}{6}}^{\frac {1}{3}}(x)+c_2 Q_{-\frac {1}{6}}^{\frac {1}{3}}(x)\right ) \\ \end{align*}