Link to actual problem [8041] \[ \boxed {16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y=0} \]
type detected by program
{"kovacic"}
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 {\left (1+x \right ) x}{4}}}{x^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{\frac {1}{4}} {\mathrm e}^{\frac {\left (1+x \right ) x}{4}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {\left (1+x \right ) x}{4}} \left (\int \frac {{\mathrm e}^{\frac {\left (1+x \right ) x}{4}}}{x}d x \right )}{x^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{4}} {\mathrm e}^{\frac {\left (1+x \right ) x}{4}} y}{\int \frac {{\mathrm e}^{\frac {\left (1+x \right ) x}{4}}}{x}d x}\right ] \\ \end{align*}