Link to actual problem [9831] \[ \boxed {2 x \left (x -1\right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+y a=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_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 {MathieuC}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {MathieuC}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {MathieuS}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {MathieuS}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {MathieuC}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \operatorname {MathieuS}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {MathieuC}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \operatorname {MathieuS}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )}\right ] \\ \end{align*}