Link to actual problem [1194] \[ \boxed {\left (3 x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
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 &= \left (3 x^{2}+1\right )^{\frac {2}{3}} \operatorname {LegendreP}\left (\frac {i \sqrt {23}}{6}-\frac {1}{2}, \frac {4}{3}, i \sqrt {3}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (3 x^{2}+1\right )^{\frac {2}{3}} \operatorname {LegendreP}\left (\frac {i \sqrt {23}}{6}-\frac {1}{2}, \frac {4}{3}, i \sqrt {3}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (3 x^{2}+1\right )^{\frac {2}{3}} \operatorname {LegendreQ}\left (\frac {i \sqrt {23}}{6}-\frac {1}{2}, \frac {4}{3}, i \sqrt {3}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (3 x^{2}+1\right )^{\frac {2}{3}} \operatorname {LegendreQ}\left (\frac {i \sqrt {23}}{6}-\frac {1}{2}, \frac {4}{3}, i \sqrt {3}\, x \right )}\right ] \\ \end{align*}