Internal problem ID [7567]
Book: Collection of Kovacic problems
Section: section 2. Solution found using all possible Kovacic cases
Problem number: 7.
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: 0.032 (sec). Leaf size: 58
dsolve((1-x^2)*diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {1}{2}-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}-\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )+c_{2} \sqrt {2 x +2}\, \hypergeom \left (\left [-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}\right ], \left [\frac {3}{2}\right ], \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 166.966 (sec). Leaf size: 371
DSolve[(1-x^2)*y''[x]+y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sqrt [4]{1-x} \left (x-\sqrt {10} \sqrt {x-1}-\sqrt {5} \sqrt {x-1} \sqrt {x+1}+\sqrt {2} \sqrt {x+1}+1\right ) e^{2 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {x+1}+\sqrt {2}}{\sqrt {x-1}}\right )} \left (c_2 \int _1^x\frac {e^{-4 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {K[1]+1}+\sqrt {2}}{\sqrt {K[1]-1}}\right )} \sqrt {\frac {K[1]-1}{K[1]+1}} \sqrt [64]{-\sqrt {K[1]-1}+\sqrt {K[1]+1}+\sqrt {2}} \sqrt [64]{\sqrt {K[1]-1}+\sqrt {K[1]+1}+\sqrt {2}} \left (\sqrt {2} \sqrt {K[1]+1}+2\right )^{127/64}}{4 \left (K[1]-\sqrt {10} \sqrt {K[1]-1}-\sqrt {5} \sqrt {K[1]-1} \sqrt {K[1]+1}+\sqrt {2} \sqrt {K[1]+1}+1\right )^2}dK[1]+c_1\right )}{\sqrt [4]{x-1} \sqrt [128]{-\sqrt {x-1}+\sqrt {x+1}+\sqrt {2}} \sqrt [128]{\sqrt {x-1}+\sqrt {x+1}+\sqrt {2}} \left (\sqrt {2} \sqrt {x+1}+2\right )^{127/128}} \\ \end{align*}