Internal problem ID [7784]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 297.
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: 0.031 (sec). Leaf size: 177
dsolve((1-x^2)*diff(y(x),x$2)-diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {2 x +3}\, {\left (\frac {3 \sqrt {5}\, x +2 \sqrt {5}-5 \sqrt {x^{2}-1}}{3 \sqrt {5}\, x +2 \sqrt {5}+5 \sqrt {x^{2}-1}}\right )}^{\frac {1}{4}} \left (x +\sqrt {x^{2}-1}\right )^{\frac {3 \sqrt {5}}{10}} \left (x +\sqrt {x^{2}-1}\right )^{\frac {\sqrt {5}}{5}}+c_{2} \sqrt {2 x +3}\, {\left (\frac {3 \sqrt {5}\, x +2 \sqrt {5}+5 \sqrt {x^{2}-1}}{3 \sqrt {5}\, x +2 \sqrt {5}-5 \sqrt {x^{2}-1}}\right )}^{\frac {1}{4}} \left (x +\sqrt {x^{2}-1}\right )^{-\frac {3 \sqrt {5}}{10}} \left (x +\sqrt {x^{2}-1}\right )^{-\frac {\sqrt {5}}{5}} \]
✓ Solution by Mathematica
Time used: 30.669 (sec). Leaf size: 198
DSolve[(1-x^2)*y''[x]-y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt [4]{x+1} \left (\sqrt {x+1}-\sqrt {x-1}\right )^{-1-\sqrt {5}} \left (-2 x+2 \sqrt {x-1} \sqrt {x+1}+\sqrt {5}-3\right ) e^{-\text {arctanh}\left (x-\sqrt {x-1} \sqrt {x+1}\right )} \left (c_2 \int _1^x\frac {e^{2 \text {arctanh}\left (K[1]-\sqrt {K[1]-1} \sqrt {K[1]+1}\right )} \left (\sqrt {K[1]+1}-\sqrt {K[1]-1}\right )^{2 \left (1+\sqrt {5}\right )}}{\left (-2 K[1]+2 \sqrt {K[1]-1} \sqrt {K[1]+1}+\sqrt {5}-3\right )^2}dK[1]+c_1\right )}{\sqrt [4]{1-x}} \]