Internal problem ID [6833]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 101.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.22 (sec). Leaf size: 41
dsolve(8*x^2*(1-x^2)*diff(y(x),x$2)+2*x*(1-13*x^2)*diff(y(x),x)+(1-9*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} x^{\frac {1}{4}}}{\sqrt {x^{2}-1}}+\frac {c_{2} x^{\frac {3}{8}} \LegendreQ \left (-\frac {1}{8}, \frac {1}{8}, \sqrt {-x^{2}+1}\right )}{\sqrt {x^{2}-1}} \]
✓ Solution by Mathematica
Time used: 10.034 (sec). Leaf size: 47
DSolve[8*x^2*(1-x^2)*y''[x]+2*x*(1-13*x^2)*y'[x]+(1-9*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [4]{x} \left (4 c_2 \sqrt [4]{x} \, _2F_1\left (\frac {1}{8},\frac {1}{2};\frac {9}{8};x^2\right )+c_1\right )}{\sqrt {1-x^2}} \\ \end{align*}