59.2.7 problem 7

Internal problem ID [10000]
Book : Collection of Kovacic problems
Section : section 2. Solution found using all possible Kovacic cases
Problem number : 7
Date solved : Monday, January 27, 2025 at 06:16:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 58

dsolve((1-x^2)*diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {hypergeom}\left (\left [\frac {\sqrt {5}}{2}-\frac {1}{2}, -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ], \left [\frac {1}{2}\right ], \frac {1}{2}+\frac {x}{2}\right )+c_{2} \sqrt {2+2 x}\, \operatorname {hypergeom}\left (\left [\frac {\sqrt {5}}{2}, -\frac {\sqrt {5}}{2}\right ], \left [\frac {3}{2}\right ], \frac {1}{2}+\frac {x}{2}\right ) \]

✓ Solution by Mathematica

Time used: 5.189 (sec). Leaf size: 215

DSolve[(1-x^2)*D[y[x],{x,2}]+D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (\sqrt {x-1}-\sqrt {x+1}\right )^{\frac {1}{2} \left (-1-\sqrt {5}\right )} \left (\sqrt {x-1}+\sqrt {x+1}\right )^{\frac {1}{2} \left (\sqrt {5}-1\right )} \left (5 \sqrt {x-1}-\sqrt {5} \sqrt {x+1}\right ) \left (c_2 \int _1^x-\frac {2 e^{-\text {arctanh}(K[2])} \left (\sqrt {K[2]-1}-\sqrt {K[2]+1}\right )^{\sqrt {5}} \left (\sqrt {K[2]-1}+\sqrt {K[2]+1}\right )^{-\sqrt {5}}}{\left (\sqrt {5} \sqrt {K[2]+1}-5 \sqrt {K[2]-1}\right )^2}dK[2]+c_1\right ) \exp \left (\frac {1}{2} \left (\text {arctanh}(x)-\int _1^x\frac {1}{1-K[1]^2}dK[1]\right )\right ) \]