Link to actual problem [9872] \[ \boxed {x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 x^{3} y^{\prime \prime }-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_high_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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= y \,{\mathrm e}^{-\frac {x^{2}}{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -{\mathrm e}^{x^{2}} \left (\int \frac {\operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right )+\left (\int \frac {\operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{-{\mathrm e}^{x^{2}} \left (\int \frac {\operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right )+\left (\int \frac {\operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -{\mathrm e}^{x^{2}} \left (\int \frac {\operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right )+\left (\int \frac {\operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{-{\mathrm e}^{x^{2}} \left (\int \frac {\operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right )+\left (\int \frac {\operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}}\right ] \\ \end{align*}