Link to actual problem [5030] \[ \boxed {y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = -1\).
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 &= \operatorname {KummerM}\left (-\frac {1}{6}, \frac {1}{2}, -\frac {3}{2} x^{2}+x -\frac {1}{6}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {KummerM}\left (-\frac {1}{6}, \frac {1}{2}, -\frac {\left (3 x -1\right )^{2}}{6}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {KummerU}\left (-\frac {1}{6}, \frac {1}{2}, -\frac {3}{2} x^{2}+x -\frac {1}{6}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {KummerU}\left (-\frac {1}{6}, \frac {1}{2}, -\frac {\left (3 x -1\right )^{2}}{6}\right )}\right ] \\ \end{align*}