Link to actual problem [9502] \[ \boxed {x^{2} y^{\prime \prime }+2 \left (x +a \right ) y^{\prime }-b \left (-1+b \right ) y=0} \]
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 {{\mathrm e}^{\frac {a}{x}} \operatorname {BesselI}\left (b -\frac {1}{2}, \frac {a}{x}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{-\frac {a}{x}} y}{\operatorname {BesselI}\left (b -\frac {1}{2}, \frac {a}{x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {a}{x}} \operatorname {BesselK}\left (b -\frac {1}{2}, \frac {a}{x}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{-\frac {a}{x}} y}{\operatorname {BesselK}\left (b -\frac {1}{2}, \frac {a}{x}\right )}\right ] \\ \end{align*}