Link to actual problem [12248] \[ \boxed {y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+y \cot \left (x \right )=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = 1, y^{\prime }\left (\frac {\pi }{4}\right ) = 0\right ] \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 &= \csc \left (x \right )^{4} \left (\cot \left (x \right )-i\right )^{-\frac {9}{4}+\frac {\sqrt {1+4 i}}{4}} \left (\cot \left (x \right )+i\right )^{-\frac {9}{4}-\frac {\sqrt {1-4 i}}{4}} \left (2 \operatorname {hypergeom}\left (\left [\frac {\sqrt {1+4 i}}{4}+\frac {1}{2}-\frac {\sqrt {1-4 i}}{4}, \frac {\sqrt {1+4 i}}{4}+\frac {1}{2}-\frac {\sqrt {1-4 i}}{4}\right ], \left [1-\frac {\sqrt {1-4 i}}{2}\right ], -\frac {i \cot \left (x \right )}{2}+\frac {1}{2}\right ) \left (-4-i+\left (2 i-\sqrt {1+4 i}\, \left (\cot \left (x \right )+i\right )-4 \cot \left (x \right )\right ) \sqrt {1-4 i}+2 \sqrt {1+4 i}\, \left (\cot \left (x \right )+i\right )+\left (5-4 i\right ) \cot \left (x \right )\right )+i \left (1+\cot \left (x \right )^{2}\right ) \left (\left (2+\sqrt {1+4 i}\right ) \sqrt {1-4 i}-2 \sqrt {1+4 i}-3\right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {1+4 i}}{4}+\frac {3}{2}-\frac {\sqrt {1-4 i}}{4}, \frac {\sqrt {1+4 i}}{4}+\frac {3}{2}-\frac {\sqrt {1-4 i}}{4}\right ], \left [2-\frac {\sqrt {1-4 i}}{2}\right ], -\frac {i \cot \left (x \right )}{2}+\frac {1}{2}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\cot \left (x \right )-i\right )^{-\frac {\sqrt {1+4 i}}{4}} \left (\cot \left (x \right )-i\right )^{\frac {9}{4}} \left (\cot \left (x \right )+i\right )^{\frac {9}{4}} \left (\cot \left (x \right )+i\right )^{\frac {\sqrt {1-4 i}}{4}} y}{8 \csc \left (x \right )^{4} \left (\operatorname {hypergeom}\left (\left [\frac {\sqrt {1+4 i}}{4}+\frac {1}{2}-\frac {\sqrt {1-4 i}}{4}, \frac {\sqrt {1+4 i}}{4}+\frac {1}{2}-\frac {\sqrt {1-4 i}}{4}\right ], \left [1-\frac {\sqrt {1-4 i}}{2}\right ], -\frac {i \cot \left (x \right )}{2}+\frac {1}{2}\right ) \left (-1-\frac {i}{4}+\left (\frac {i}{2}-\frac {\sqrt {1+4 i}\, \left (\cot \left (x \right )+i\right )}{4}-\cot \left (x \right )\right ) \sqrt {1-4 i}+\frac {\sqrt {1+4 i}\, \left (\cot \left (x \right )+i\right )}{2}+\left (\frac {5}{4}-i\right ) \cot \left (x \right )\right )+\frac {i \left (1+\cot \left (x \right )^{2}\right ) \left (\left (2+\sqrt {1+4 i}\right ) \sqrt {1-4 i}-2 \sqrt {1+4 i}-3\right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {1+4 i}}{4}+\frac {3}{2}-\frac {\sqrt {1-4 i}}{4}, \frac {\sqrt {1+4 i}}{4}+\frac {3}{2}-\frac {\sqrt {1-4 i}}{4}\right ], \left [2-\frac {\sqrt {1-4 i}}{2}\right ], -\frac {i \cot \left (x \right )}{2}+\frac {1}{2}\right )}{8}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\left (\left (i \sqrt {1+4 i}\, \cot \left (x \right )+4 i \cot \left (x \right )-\sqrt {1+4 i}+2\right ) \sqrt {1-4 i}+\left (2 i \cot \left (x \right )-2\right ) \sqrt {1+4 i}+1-4 i+\left (4+5 i\right ) \cot \left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {1+4 i}}{4}+\frac {1}{2}+\frac {\sqrt {1-4 i}}{4}, \frac {\sqrt {1+4 i}}{4}+\frac {1}{2}+\frac {\sqrt {1-4 i}}{4}\right ], \left [1+\frac {\sqrt {1-4 i}}{2}\right ], -\frac {i \cot \left (x \right )}{2}+\frac {1}{2}\right )+\frac {\left (1+\cot \left (x \right )^{2}\right ) \left (\left (2+\sqrt {1+4 i}\right ) \sqrt {1-4 i}+2 \sqrt {1+4 i}+3\right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {1+4 i}}{4}+\frac {3}{2}+\frac {\sqrt {1-4 i}}{4}, \frac {\sqrt {1+4 i}}{4}+\frac {3}{2}+\frac {\sqrt {1-4 i}}{4}\right ], \left [2+\frac {\sqrt {1-4 i}}{2}\right ], -\frac {i \cot \left (x \right )}{2}+\frac {1}{2}\right )}{2}\right ) \csc \left (x \right )^{4} \left (\cot \left (x \right )-i\right )^{-\frac {9}{4}+\frac {\sqrt {1+4 i}}{4}} \left (\cot \left (x \right )+i\right )^{-\frac {9}{4}+\frac {\sqrt {1-4 i}}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\cot \left (x \right )-i\right )^{\frac {9}{4}} \left (\cot \left (x \right )-i\right )^{-\frac {\sqrt {1+4 i}}{4}} \left (\cot \left (x \right )+i\right )^{\frac {9}{4}} \left (\cot \left (x \right )+i\right )^{-\frac {\sqrt {1-4 i}}{4}} y}{\left (\left (\left (i \sqrt {1+4 i}\, \cot \left (x \right )+4 i \cot \left (x \right )-\sqrt {1+4 i}+2\right ) \sqrt {1-4 i}+\left (2 i \cot \left (x \right )-2\right ) \sqrt {1+4 i}+1-4 i+\left (4+5 i\right ) \cot \left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {1+4 i}}{4}+\frac {1}{2}+\frac {\sqrt {1-4 i}}{4}, \frac {\sqrt {1+4 i}}{4}+\frac {1}{2}+\frac {\sqrt {1-4 i}}{4}\right ], \left [1+\frac {\sqrt {1-4 i}}{2}\right ], -\frac {i \cot \left (x \right )}{2}+\frac {1}{2}\right )+\frac {\left (1+\cot \left (x \right )^{2}\right ) \left (\left (2+\sqrt {1+4 i}\right ) \sqrt {1-4 i}+2 \sqrt {1+4 i}+3\right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {1+4 i}}{4}+\frac {3}{2}+\frac {\sqrt {1-4 i}}{4}, \frac {\sqrt {1+4 i}}{4}+\frac {3}{2}+\frac {\sqrt {1-4 i}}{4}\right ], \left [2+\frac {\sqrt {1-4 i}}{2}\right ], -\frac {i \cot \left (x \right )}{2}+\frac {1}{2}\right )}{2}\right ) \csc \left (x \right )^{4}}\right ] \\ \end{align*}