Link to actual problem [9851] \[ \boxed {\left (-a +x \right )^{3} \left (x -b \right )^{3} y^{\prime \prime \prime }-c y=0} \]
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 &= \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=1\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=1\right )}{a -b}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (x -a \right )^{\frac {2 b}{a -b}} \left (x -b \right )^{-\frac {2 a}{a -b}} \left (-x +b \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=1\right )}{a -b}} \left (-x +a \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=1\right )}{a -b}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=2\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=2\right )}{a -b}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (x -a \right )^{\frac {2 b}{a -b}} \left (x -b \right )^{-\frac {2 a}{a -b}} \left (-x +b \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=2\right )}{a -b}} \left (-x +a \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=2\right )}{a -b}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=3\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=3\right )}{a -b}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (x -a \right )^{\frac {2 b}{a -b}} \left (x -b \right )^{-\frac {2 a}{a -b}} \left (-x +b \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=3\right )}{a -b}} \left (-x +a \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} &=3\right )}{a -b}} y\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= y \left (x -a \right )^{-\frac {2 a}{a -b}} \left (x -b \right )^{\frac {2 b}{a -b}}, S \left (R \right ) &= \frac {\ln \left (x -a \right )-\ln \left (x -b \right )}{a -b}\right ] \\ \end{align*}