Link to actual problem [14804] \[ \boxed {\left (x^{2}-25\right )^{2} y^{\prime \prime }-\left (x +5\right ) y^{\prime }+10 y=0} \] 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 &= \left (x +5\right )^{\frac {101}{200}+\frac {3 \sqrt {689}}{200}} \left (x -5\right )^{\frac {99}{200}-\frac {3 \sqrt {689}}{200}} {\mathrm e}^{-\frac {15+x}{200 \left (x -5\right )}} \operatorname {KummerM}\left (\frac {299}{200}+\frac {3 \sqrt {689}}{200}, 1+\frac {3 \sqrt {689}}{100}, \frac {x +5}{100 x -500}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +5\right )^{-\frac {3 \sqrt {689}}{200}} \left (x -5\right )^{\frac {3 \sqrt {689}}{200}} {\mathrm e}^{\frac {15+x}{200 x -1000}} y}{\left (x +5\right )^{\frac {101}{200}} \left (x -5\right )^{\frac {99}{200}} \operatorname {KummerM}\left (\frac {299}{200}+\frac {3 \sqrt {689}}{200}, 1+\frac {3 \sqrt {689}}{100}, \frac {x +5}{100 x -500}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x +5\right )^{\frac {101}{200}+\frac {3 \sqrt {689}}{200}} \left (x -5\right )^{\frac {99}{200}-\frac {3 \sqrt {689}}{200}} {\mathrm e}^{-\frac {15+x}{200 \left (x -5\right )}} \operatorname {KummerU}\left (\frac {299}{200}+\frac {3 \sqrt {689}}{200}, 1+\frac {3 \sqrt {689}}{100}, \frac {x +5}{100 x -500}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +5\right )^{-\frac {3 \sqrt {689}}{200}} \left (x -5\right )^{\frac {3 \sqrt {689}}{200}} {\mathrm e}^{\frac {15+x}{200 x -1000}} y}{\left (x +5\right )^{\frac {101}{200}} \left (x -5\right )^{\frac {99}{200}} \operatorname {KummerU}\left (\frac {299}{200}+\frac {3 \sqrt {689}}{200}, 1+\frac {3 \sqrt {689}}{100}, \frac {x +5}{100 x -500}\right )}\right ] \\ \end{align*}