Link to actual problem [9773] \[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {\left (-x -1\right ) y}{x^{4}}=0} \]
type detected by program
{"kovacic", "second_order_bessel_ode"}
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 &= {\mathrm e}^{\frac {1}{x}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {1}{x}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {1}{x}} \operatorname {expIntegral}_{1}\left (\frac {2}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{x}} y}{\operatorname {expIntegral}_{1}\left (\frac {2}{x}\right )}\right ] \\ \end{align*}