Internal problem ID [8331]
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]]
\[ \boxed {y^{\prime \prime }+\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 103
dsolve(diff(y(x),x$2)= -(5*x^2+27)/(36*(x^2-1)^2)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x^{2}-1\right )^{\frac {1}{3}} {\mathrm e}^{\int \operatorname {RootOf}\left (-1+\left (432 x^{4}-864 x^{2}+432\right ) \textit {\_Z}^{4}+\left (-72 x^{2}+72\right ) \textit {\_Z}^{2}+16 x \textit {\_Z} , \operatorname {index} =1\right )d x}+c_{2} \left (x^{2}-1\right )^{\frac {1}{3}} {\mathrm e}^{\int \operatorname {RootOf}\left (-1+\left (432 x^{4}-864 x^{2}+432\right ) \textit {\_Z}^{4}+\left (-72 x^{2}+72\right ) \textit {\_Z}^{2}+16 x \textit {\_Z} , \operatorname {index} =2\right )d x} \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 38
DSolve[y''[x]== -(5*x^2+27)/(36*(x^2-1)^2)*y[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {x^2-1} \left (c_1 P_{-\frac {1}{6}}^{\frac {1}{3}}(x)+c_2 Q_{-\frac {1}{6}}^{\frac {1}{3}}(x)\right ) \]