Link to actual problem [9762] \[ \boxed {y^{\prime \prime }+\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}}=0} \]
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 &= \sqrt {\sin \left (x \right )}\, \operatorname {LegendreP}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {\sin \left (x \right )}\, \operatorname {LegendreP}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {\sin \left (x \right )}\, \operatorname {LegendreQ}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {\sin \left (x \right )}\, \operatorname {LegendreQ}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )}\right ] \\ \end{align*}