Link to actual problem [7020] \[ \boxed {4 x^{2} y^{\prime \prime }+3 y^{\prime } x^{2}+\left (1+3 x \right ) y=0} \] 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 &= {\mathrm e}^{-\frac {3 x}{8}} \sqrt {x}\, \left (3 \operatorname {BesselI}\left (0, \frac {3 x}{8}\right ) x -3 x \operatorname {BesselI}\left (1, \frac {3 x}{8}\right )-4 \operatorname {BesselI}\left (0, \frac {3 x}{8}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {3 x}{8}} y}{\sqrt {x}\, \left (\left (3 x -4\right ) \operatorname {BesselI}\left (0, \frac {3 x}{8}\right )-3 x \operatorname {BesselI}\left (1, \frac {3 x}{8}\right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {3 x}{8}} \sqrt {x}\, \left (3 \operatorname {BesselK}\left (0, -\frac {3 x}{8}\right ) x -3 x \operatorname {BesselK}\left (1, -\frac {3 x}{8}\right )-4 \operatorname {BesselK}\left (0, -\frac {3 x}{8}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {3 x}{8}} y}{\sqrt {x}\, \left (\left (3 x -4\right ) \operatorname {BesselK}\left (0, -\frac {3 x}{8}\right )-3 x \operatorname {BesselK}\left (1, -\frac {3 x}{8}\right )\right )}\right ] \\ \end{align*}