Link to actual problem [9671] \[ \boxed {y^{\prime \prime }+\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}=0} \]
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 &= x \operatorname {BesselJ}\left (v , {\mathrm e}^{\frac {1}{x}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselJ}\left (v , {\mathrm e}^{\frac {1}{x}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {BesselY}\left (v , {\mathrm e}^{\frac {1}{x}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselY}\left (v , {\mathrm e}^{\frac {1}{x}}\right )}\right ] \\ \end{align*}