Link to actual problem [14785] \[ \boxed {\left (3 x +1\right ) y^{\prime \prime }-3 y^{\prime }-2 y=0} \] 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 &= \frac {\operatorname {BesselI}\left (2, \frac {2 \sqrt {6 x +2}}{3}\right ) \left (1+3 x \right )}{3}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {3 y}{\operatorname {BesselI}\left (2, \frac {2 \sqrt {6 x +2}}{3}\right ) \left (1+3 x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {BesselY}\left (2, \frac {2 i \sqrt {6 x +2}}{3}\right ) \left (1+3 x \right )}{3}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {3 y}{\operatorname {BesselY}\left (2, \frac {2 i \sqrt {6 x +2}}{3}\right ) \left (1+3 x \right )}\right ] \\ \end{align*}