Link to actual problem [9392] \[ \boxed {y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y=0} \]
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 &= x \,{\mathrm e}^{-\frac {a \,x^{q}}{q}} \operatorname {KummerM}\left (1-\frac {b}{a q}, 1+\frac {1}{q}, \frac {a \,x^{q}}{q}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a \,x^{q}}{q}} y}{x \operatorname {KummerM}\left (\frac {a q -b}{a q}, \frac {q +1}{q}, \frac {a \,x^{q}}{q}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \,{\mathrm e}^{-\frac {a \,x^{q}}{q}} \operatorname {KummerU}\left (1-\frac {b}{a q}, 1+\frac {1}{q}, \frac {a \,x^{q}}{q}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a \,x^{q}}{q}} y}{x \operatorname {KummerU}\left (\frac {a q -b}{a q}, \frac {q +1}{q}, \frac {a \,x^{q}}{q}\right )}\right ] \\ \end{align*}