Internal problem ID [7460]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 741.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 62
dsolve((1-x^2)*diff(y(x),x$2)-diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {1}{2}-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}-\frac {1}{2}\right ], \left [-\frac {1}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )+c_{2} \left (2 x +2\right )^{\frac {3}{2}} \hypergeom \left (\left [1-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}+1\right ], \left [\frac {5}{2}\right ], \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 1.422 (sec). Leaf size: 219
DSolve[(1-x^2)*y''[x]-y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [4]{x+1} \left (\sqrt {x-1}-\sqrt {x+1}\right )^{-1-\sqrt {5}} \left (-2 x+2 \sqrt {x-1} \sqrt {x+1}+\sqrt {5}-3\right ) e^{-\tanh ^{-1}\left (x-\sqrt {x-1} \sqrt {x+1}\right )} \left (c_2 \int _1^x-\frac {2 e^{2 \tanh ^{-1}\left (K[1]-\sqrt {K[1]-1} \sqrt {K[1]+1}\right )} \left (\sqrt {K[1]-1}-\sqrt {K[1]+1}\right )^{2 \sqrt {5}} \left (\sqrt {K[1]-1} \sqrt {K[1]+1}-K[1]\right )}{\left (-2 K[1]+2 \sqrt {K[1]-1} \sqrt {K[1]+1}+\sqrt {5}-3\right )^2}dK[1]+c_1\right )}{\sqrt [4]{1-x}} \\ \end{align*}