Link to actual problem [9354] \[ \boxed {y^{\prime \prime }+\left (a \cosh \left (x \right )^{2}+b \right ) y=0} \]
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 {MathieuC}\left (-\frac {a}{2}-b , \frac {a}{4}, i x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {MathieuC}\left (-\frac {a}{2}-b , \frac {a}{4}, i x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {MathieuS}\left (-\frac {a}{2}-b , \frac {a}{4}, i x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {MathieuS}\left (-\frac {a}{2}-b , \frac {a}{4}, i x \right )}\right ] \\ \end{align*}