Link to actual problem [6042] \[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime }+3 y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Irregular singular point"}
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 &= \sqrt {x}\, \operatorname {HeunD}\left (\left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}, -1+\left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}-20 i \sqrt {3}, \left (-8+8 i\right ) 3^{\frac {1}{4}} \sqrt {5}, \left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}+1+20 i \sqrt {3}, \frac {5+\left (-1+i\right ) 3^{\frac {1}{4}} \sqrt {5}\, x}{-5+\left (-1+i\right ) 3^{\frac {1}{4}} \sqrt {5}\, x}\right ) {\mathrm e}^{i \sqrt {3}\, x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-i \sqrt {3}\, x} y}{\sqrt {x}\, \operatorname {HeunD}\left (\left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}, -1+\left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}-20 i \sqrt {3}, \left (-8+8 i\right ) 3^{\frac {1}{4}} \sqrt {5}, \left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}+1+20 i \sqrt {3}, \frac {5+\left (-1+i\right ) 3^{\frac {1}{4}} \sqrt {5}\, x}{-5+\left (-1+i\right ) 3^{\frac {1}{4}} \sqrt {5}\, x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \operatorname {HeunD}\left (\left (-4+4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}, -1+\left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}-20 i \sqrt {3}, \left (-8+8 i\right ) 3^{\frac {1}{4}} \sqrt {5}, \left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}+1+20 i \sqrt {3}, \frac {5+\left (-1+i\right ) 3^{\frac {1}{4}} \sqrt {5}\, x}{-5+\left (-1+i\right ) 3^{\frac {1}{4}} \sqrt {5}\, x}\right ) {\mathrm e}^{-i \sqrt {3}\, x -\frac {5}{x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{i \sqrt {3}\, x} {\mathrm e}^{\frac {5}{x}} y}{\sqrt {x}\, \operatorname {HeunD}\left (\left (-4+4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}, -1+\left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}-20 i \sqrt {3}, \left (-8+8 i\right ) 3^{\frac {1}{4}} \sqrt {5}, \left (4-4 i\right ) \sqrt {5}\, 3^{\frac {1}{4}}+1+20 i \sqrt {3}, \frac {5+\left (-1+i\right ) 3^{\frac {1}{4}} \sqrt {5}\, x}{-5+\left (-1+i\right ) 3^{\frac {1}{4}} \sqrt {5}\, x}\right )}\right ] \\ \end{align*}