Link to actual problem [9471] \[ \boxed {5 \left (x a +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (x a +b \right )^{{1}/{5}} y=0} \]
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 &= \frac {\sinh \left (\frac {\left (x a +b \right )^{{3}/{5}} \sqrt {-5 c}}{3 a}\right )}{\left (x a +b \right )^{{3}/{5}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x a +b \right )^{{3}/{5}} y}{\sinh \left (\frac {\left (x a +b \right )^{{3}/{5}} \sqrt {-5 c}}{3 a}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\cosh \left (\frac {\left (x a +b \right )^{{3}/{5}} \sqrt {-5 c}}{3 a}\right )}{\left (x a +b \right )^{{3}/{5}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x a +b \right )^{{3}/{5}} y}{\cosh \left (\frac {\left (x a +b \right )^{{3}/{5}} \sqrt {-5 c}}{3 a}\right )}\right ] \\ \end{align*}