59.3.6 problem Kovacic 1985 paper. page 25. section 5.2. Example 2

Internal problem ID [10008]
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
Date solved : Monday, January 27, 2025 at 06:16:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 25

dsolve(diff(y(x),x$2)= -(5*x^2+27)/(36*(x^2-1)^2)*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 38

DSolve[D[y[x],{x,2}]== -(5*x^2+27)/(36*(x^2-1)^2)*y[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {x^2-1} \left (c_1 P_{-\frac {1}{6}}^{\frac {1}{3}}(x)+c_2 Q_{-\frac {1}{6}}^{\frac {1}{3}}(x)\right ) \]