Internal problem ID [7258]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 538.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 53
dsolve(4*x^2*(1-x^2)*diff(y(x),x$2)+x*(7-19*x^2)*diff(y(x),x)-(1+14*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \LegendreP \left (-\frac {3}{8}, \frac {5}{8}, \sqrt {-x^{2}+1}\right )}{x^{\frac {3}{8}} \sqrt {x^{2}-1}}+\frac {c_{2} \LegendreQ \left (-\frac {3}{8}, \frac {5}{8}, \sqrt {-x^{2}+1}\right )}{x^{\frac {3}{8}} \sqrt {x^{2}-1}} \]
✓ Solution by Mathematica
Time used: 10.063 (sec). Leaf size: 50
DSolve[4*x^2*(1-x^2)*y''[x]+x*(7-19*x^2)*y'[x]-(1+14*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {4 c_2 x^{5/4} \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};x^2\right )+5 c_1}{5 x \sqrt {1-x^2}} \\ \end{align*}