Link to actual problem [12249] \[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } \left (x -1\right )+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x +i\right )^{\frac {1}{2}+\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i}{2}, \frac {1}{2}+\frac {3 i}{2}\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +i\right )^{-\frac {1}{2}-\frac {i}{2}} y}{\operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i}{2}, \frac {1}{2}+\frac {3 i}{2}\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], -\frac {i x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}