Link to actual problem [11014] \[ \boxed {x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +\lambda b \right ) y^{\prime }+\lambda \left (-2 a +c \right ) y=0} \]
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 {\left (x a +b \right )^{3-\frac {c}{a}} \operatorname {HeunC}\left (\frac {\lambda a}{b}, \frac {c}{a}-1, 3-\frac {c}{a}, 0, \frac {5}{2}-\frac {2 a^{3} \lambda -a^{2} c \lambda +4 a b c -b \,c^{2}}{2 a^{2} b}, -\frac {b}{a x}\right )}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x a +b \right )^{\frac {c}{a}} x^{2} y}{\left (x a +b \right )^{3} \operatorname {HeunC}\left (\frac {\lambda a}{b}, \frac {-a +c}{a}, \frac {-c +3 a}{a}, 0, \frac {-2 a^{3} \lambda +\left (c \lambda +5 b \right ) a^{2}-4 a b c +b \,c^{2}}{2 a^{2} b}, -\frac {b}{a x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x a +b \right )^{3-\frac {c}{a}} x^{-3+\frac {c}{a}} \operatorname {HeunC}\left (\frac {\lambda a}{b}, 1-\frac {c}{a}, 3-\frac {c}{a}, 0, \frac {5}{2}-\frac {2 a^{3} \lambda -a^{2} c \lambda +4 a b c -b \,c^{2}}{2 a^{2} b}, -\frac {b}{a x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x a +b \right )^{\frac {c}{a}} x^{3} x^{-\frac {c}{a}} y}{\left (x a +b \right )^{3} \operatorname {HeunC}\left (\frac {\lambda a}{b}, \frac {a -c}{a}, \frac {-c +3 a}{a}, 0, \frac {-2 a^{3} \lambda +\left (c \lambda +5 b \right ) a^{2}-4 a b c +b \,c^{2}}{2 a^{2} b}, -\frac {b}{a x}\right )}\right ] \\ \end{align*}