Link to actual problem [6492] Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
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 &= \operatorname {hypergeom}\left (\left [1\right ], \left [\frac {1}{2}-\frac {i \sqrt {3}}{2}, \frac {1}{2}+\frac {i \sqrt {3}}{2}\right ], -x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [1\right ], \left [\frac {1}{2}-\frac {i \sqrt {3}}{2}, \frac {1}{2}+\frac {i \sqrt {3}}{2}\right ], -x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \operatorname {BesselI}\left (-i \sqrt {3}, 2 \sqrt {-x}\right ) \left (-x \right )^{\frac {i \sqrt {3}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {i \sqrt {3}}{2}} \left (-x \right )^{-\frac {i \sqrt {3}}{2}} y}{\sqrt {x}\, \operatorname {BesselI}\left (-i \sqrt {3}, 2 \sqrt {-x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \operatorname {BesselI}\left (i \sqrt {3}, 2 \sqrt {-x}\right ) \left (-x \right )^{-\frac {i \sqrt {3}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {i \sqrt {3}}{2}} \left (-x \right )^{\frac {i \sqrt {3}}{2}} y}{\sqrt {x}\, \operatorname {BesselI}\left (i \sqrt {3}, 2 \sqrt {-x}\right )}\right ] \\ \end{align*}