Link to actual problem [10849] \[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (a x +b -c \right ) y=0} \]
type detected by program
{"kovacic"}
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-c x} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x a +b -2 c \right )}{2 \sqrt {a}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{c x} y}{\operatorname {erf}\left (\frac {\sqrt {2}\, \left (x a +b -2 c \right )}{2 \sqrt {a}}\right )}\right ] \\ \end{align*}