Link to actual problem [9828] \[ \boxed {x^{2} y^{\prime \prime \prime }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y=0} \]
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 &= \frac {1}{x^{2}}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselK}\left (\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+2 \operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselK}\left (\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+2 \operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x}\right ] \\ \end{align*}