Link to actual problem [10852] \[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y=0} \]
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 &= {\mathrm e}^{-\frac {c \,x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {c \,x^{2}}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {c \,x^{2}}{2}} \operatorname {erf}\left (\frac {x a -2 c x +b}{\sqrt {2 a -4 c}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {c \,x^{2}}{2}} y}{\operatorname {erf}\left (\frac {\left (-2 c +a \right ) x +b}{\sqrt {2 a -4 c}}\right )}\right ] \\ \end{align*}