Link to actual problem [9662] \[ \boxed {y^{\prime \prime }+\frac {\left (-1+3 x \right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {\left (x a +b \right ) y}{4 x \left (x -1\right )^{2}}=0} \]
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 &= \operatorname {LegendreP}\left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {LegendreP}\left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {LegendreQ}\left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {LegendreQ}\left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right )}\right ] \\ \end{align*}