Link to actual problem [6636] \[ \boxed {9 x^{2} y^{\prime \prime }+9 y^{\prime } x +\left (x^{6}-36\right ) y=0} \]
type detected by program
{"second_order_bessel_ode"}
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 {BesselJ}\left (\frac {2}{3}, \frac {x^{3}}{9}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselJ}\left (\frac {2}{3}, \frac {x^{3}}{9}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {BesselY}\left (\frac {2}{3}, \frac {x^{3}}{9}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselY}\left (\frac {2}{3}, \frac {x^{3}}{9}\right )}\right ] \\ \end{align*}