59.1.294 problem 297

Internal problem ID [9466]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 297
Date solved : Monday, January 27, 2025 at 06:03:07 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: 0.027 (sec). Leaf size: 66

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

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

✓ Solution by Mathematica

Time used: 6.100 (sec). Leaf size: 210

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

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