Link to actual problem [9360] \[ \boxed {y^{\prime \prime }-\left (n \left (1+n \right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y=0} \]
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 &= \operatorname {HeunG}\left (\frac {1}{k^{2}}, \frac {b}{4 k^{2}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \operatorname {JacobiSN}\left (x , k\right )^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunG}\left (\frac {1}{k^{2}}, \frac {b}{4 k^{2}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \operatorname {JacobiSN}\left (x , k\right )^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunG}\left (\frac {1}{k^{2}}, \frac {1}{4}+\frac {b +1}{4 k^{2}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \operatorname {JacobiSN}\left (x , k\right )^{2}\right ) \operatorname {JacobiSN}\left (x , k\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunG}\left (\frac {1}{k^{2}}, \frac {k^{2}+b +1}{4 k^{2}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \operatorname {JacobiSN}\left (x , k\right )^{2}\right ) \operatorname {JacobiSN}\left (x , k\right )}\right ] \\ \end{align*}