Link to actual problem [1358] \[ \boxed {9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y=0} \] 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 &= \frac {{\mathrm e}^{\frac {1}{3} x^{2}-x}}{x^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{3}} {\mathrm e}^{x} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {1}{3} x^{2}-x} \left (\int \frac {{\mathrm e}^{-\frac {1}{3} x^{2}+x}}{x}d x \right )}{x^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{3}} {\mathrm e}^{x} y}{\int \frac {{\mathrm e}^{-\frac {x^{2}}{3}} {\mathrm e}^{x}}{x}d x}\right ] \\ \end{align*}