Link to actual problem [9848] \[ \boxed {x^{6} y^{\prime \prime \prime }+y^{\prime \prime } x^{2}-2 y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{{11}/{2}}}d x \right ) x^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{{11}/{2}}}d x \right ) x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\operatorname {BesselK}\left (\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{{11}/{2}}}d x \right ) x^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\operatorname {BesselK}\left (\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{{11}/{2}}}d x \right ) x^{2}}\right ] \\ \end{align*}