Link to actual problem [9842] \[ \boxed {\left (x^{2}+1\right ) x y^{\prime \prime \prime }+3 \left (2 x^{2}+1\right ) y^{\prime \prime }-12 y=0} \]
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 &= x^{2}+\frac {1}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2}+\frac {1}{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x^{2}+1}\, x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x^{2}+1}\, x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {3 \sqrt {x^{2}+1}\, x^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )-3 x^{2}-1}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{3 \sqrt {x^{2}+1}\, x^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )-3 x^{2}-1}\right ] \\ \end{align*}