Link to actual problem [5813] \[ \boxed {y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+y \cos \left (x \right )=0} \]
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 &= \left (\cos \left (x \right )+1\right ) \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (\cos \left (x \right )+1\right ) \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\cos \left (x \right )}{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 (\cos \left (x \right )+1\right ) \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \left (\int _{}^{\cos \left (x \right )}\frac {1}{\left (\textit {\_a} +1\right )^{2} \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\textit {\_a}}{2}+\frac {1}{2}\right )^{2}}d \textit {\_a} \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (\cos \left (x \right )+1\right ) \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \left (\int _{}^{\cos \left (x \right )}\frac {1}{\left (\textit {\_a} +1\right )^{2} \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\textit {\_a}}{2}+\frac {1}{2}\right )^{2}}d \textit {\_a} \right )}\right ] \\ \end{align*}