Link to actual problem [1388] \[ \boxed {4 x^{2} \left (4 x^{2}+1\right ) y^{\prime \prime }+32 y^{\prime } x^{3}+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
type detected by Maple
[[_2nd_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 {LegendreP}\left (-\frac {1}{4}, 8 x^{2}+1\right ) \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {LegendreP}\left (-\frac {1}{4}, 8 x^{2}+1\right ) \sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {LegendreQ}\left (-\frac {3}{4}, 8 x^{2}+1\right ) \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {LegendreQ}\left (-\frac {3}{4}, 8 x^{2}+1\right ) \sqrt {x}}\right ] \\ \end{align*}