Link to actual problem [6494] Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }-2 x^{2} y^{\prime \prime }+\left (x^{2}+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 [R &= x, S \left (R \right ) &= \frac {y}{2+x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{{3}/{2}} \operatorname {BesselJ}\left (3, 2 \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{{3}/{2}} \operatorname {BesselJ}\left (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^{{3}/{2}} \operatorname {BesselY}\left (3, 2 \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{{3}/{2}} \operatorname {BesselY}\left (3, 2 \sqrt {x}\right )}\right ] \\ \end{align*}