Link to actual problem [6495] Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }+\left (2 x^{3}-x^{2}\right ) y^{\prime \prime }-y^{\prime } x +y=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*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{{3}/{2}} {\mathrm e}^{-x} \operatorname {BesselI}\left (\frac {\sqrt {13}}{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{x^{{3}/{2}} \operatorname {BesselI}\left (\frac {\sqrt {13}}{2}, x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{{3}/{2}} {\mathrm e}^{-x} \operatorname {BesselK}\left (\frac {\sqrt {13}}{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{x^{{3}/{2}} \operatorname {BesselK}\left (\frac {\sqrt {13}}{2}, x\right )}\right ] \\ \end{align*}