Link to actual problem [6586] \[ \boxed {\left (x^{2}-9\right )^{2} y^{\prime \prime }+\left (x +3\right ) y^{\prime }+2 y=0} \] 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 +3\right )^{\frac {35}{72}+\frac {\sqrt {937}}{72}} \left (x -3\right )^{\frac {37}{72}-\frac {\sqrt {937}}{72}} \operatorname {KummerM}\left (-\frac {37}{72}+\frac {\sqrt {937}}{72}, 1+\frac {\sqrt {937}}{36}, \frac {x +3}{36 x -108}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +3\right )^{-\frac {\sqrt {937}}{72}} \left (x -3\right )^{\frac {\sqrt {937}}{72}} y}{\left (x +3\right )^{\frac {35}{72}} \left (x -3\right )^{\frac {37}{72}} \operatorname {KummerM}\left (-\frac {37}{72}+\frac {\sqrt {937}}{72}, 1+\frac {\sqrt {937}}{36}, \frac {x +3}{36 x -108}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x +3\right )^{\frac {35}{72}+\frac {\sqrt {937}}{72}} \left (x -3\right )^{\frac {37}{72}-\frac {\sqrt {937}}{72}} \operatorname {KummerU}\left (-\frac {37}{72}+\frac {\sqrt {937}}{72}, 1+\frac {\sqrt {937}}{36}, \frac {x +3}{36 x -108}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +3\right )^{-\frac {\sqrt {937}}{72}} \left (x -3\right )^{\frac {\sqrt {937}}{72}} y}{\left (x +3\right )^{\frac {35}{72}} \left (x -3\right )^{\frac {37}{72}} \operatorname {KummerU}\left (-\frac {37}{72}+\frac {\sqrt {937}}{72}, 1+\frac {\sqrt {937}}{36}, \frac {x +3}{36 x -108}\right )}\right ] \\ \end{align*}