Link to actual problem [9782] \[ \boxed {y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_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 (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right ) \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right ) \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )}\right ] \\ \end{align*}