Link to actual problem [1371] \[ \boxed {x^{2} y^{\prime \prime }-x \left (3-2 x \right ) y^{\prime }+\left (3 x +4\right ) y=0} \] 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 &= \left (-x^{5}+3 x^{4}-\frac {23}{16} x^{3}\right ) {\mathrm e}^{-x} \operatorname {BesselK}\left (1, -x \right )+\left (x^{5}-\frac {7}{2} x^{4}+\frac {45}{16} x^{3}-\frac {15}{32} x^{2}\right ) {\mathrm e}^{-x} \operatorname {BesselK}\left (0, -x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\left (\left (x^{3}-\frac {7}{2} x^{2}+\frac {45}{16} x -\frac {15}{32}\right ) \operatorname {BesselK}\left (0, -x \right )-\operatorname {BesselK}\left (1, -x \right ) \left (x^{2}-3 x +\frac {23}{16}\right ) x \right ) x^{2}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{6 \left (\left (\frac {13}{3} x^{3}-\frac {101}{6} x^{2}+15 x -\frac {5}{2}\right ) \operatorname {BesselI}\left (0, x\right )+x \left (-\left (\frac {31}{6} x^{2}-18 x +\frac {34}{3}\right ) \operatorname {BesselI}\left (1, x\right )+\left (-\frac {1}{6} x^{2}-2 x +\frac {11}{3}\right ) \operatorname {BesselI}\left (1, x\right )+\operatorname {BesselI}\left (0, x\right ) x \left (x -\frac {11}{6}\right )\right )\right ) x^{2}}\right ] \\ \end{align*}