Internal problem ID [8323]
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]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 58
dsolve((1-x^2)*diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}-\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )+c_{2} \sqrt {2 x +2}\, \operatorname {hypergeom}\left (\left [\frac {\sqrt {5}}{2}, -\frac {\sqrt {5}}{2}\right ], \left [\frac {3}{2}\right ], \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 36.335 (sec). Leaf size: 171
DSolve[(1-x^2)*y''[x]+y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {\sqrt [4]{1-x} \left (\sqrt {5} \sqrt {x-1}-\sqrt {x+1}\right ) e^{2 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {x+1}+\sqrt {2}}{\sqrt {x-1}}\right )} \left (c_2 \int _1^x\frac {2 e^{-4 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {K[1]+1}+\sqrt {2}}{\sqrt {K[1]-1}}\right )} \sqrt {\frac {K[1]-1}{K[1]+1}}}{\left (\sqrt {K[1]+1}-\sqrt {5} \sqrt {K[1]-1}\right )^2}dK[1]+c_1\right )}{\sqrt {2} \sqrt [4]{x-1}} \]