Link to actual problem [9402] \[ \boxed {y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+y b=0} \]
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 &= \cos \left (x \right )^{\frac {1}{2}+\frac {a}{2}} \operatorname {LegendreP}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \left (x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\cos \left (x \right )^{-\frac {a}{2}} y}{\sqrt {\cos \left (x \right )}\, \operatorname {LegendreP}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \cos \left (x \right )^{\frac {1}{2}+\frac {a}{2}} \operatorname {LegendreQ}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \left (x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\cos \left (x \right )^{-\frac {a}{2}} y}{\sqrt {\cos \left (x \right )}\, \operatorname {LegendreQ}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \left (x \right )\right )}\right ] \\ \end{align*}