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]]
\[ \boxed {\left (1-x^{2}\right ) y^{\prime \prime }+y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 177
dsolve((1-x^2)*diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {-3+2 x}\, {\left (\frac {3 \sqrt {5}\, x -2 \sqrt {5}-5 \sqrt {x^{2}-1}}{3 \sqrt {5}\, x -2 \sqrt {5}+5 \sqrt {x^{2}-1}}\right )}^{\frac {1}{4}} \left (x +\sqrt {x^{2}-1}\right )^{\frac {3 \sqrt {5}}{10}} \left (x +\sqrt {x^{2}-1}\right )^{\frac {\sqrt {5}}{5}}+c_{2} \sqrt {-3+2 x}\, {\left (\frac {3 \sqrt {5}\, x -2 \sqrt {5}+5 \sqrt {x^{2}-1}}{3 \sqrt {5}\, x -2 \sqrt {5}-5 \sqrt {x^{2}-1}}\right )}^{\frac {1}{4}} \left (x +\sqrt {x^{2}-1}\right )^{-\frac {3 \sqrt {5}}{10}} \left (x +\sqrt {x^{2}-1}\right )^{-\frac {\sqrt {5}}{5}} \]
✓ 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]
\[ y(x)\to -\frac {\sqrt [4]{1-x} \left (\sqrt {5} \sqrt {x-1}-\sqrt {x+1}\right ) e^{2 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {x+1}+\sqrt {2}}{\sqrt {x-1}}\right )} \left (c_2 \int _1^x\frac {2 e^{-4 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {K[1]+1}+\sqrt {2}}{\sqrt {K[1]-1}}\right )} \sqrt {\frac {K[1]-1}{K[1]+1}}}{\left (\sqrt {K[1]+1}-\sqrt {5} \sqrt {K[1]-1}\right )^2}dK[1]+c_1\right )}{\sqrt {2} \sqrt [4]{x-1}} \]