Link to actual problem [9608] \[ \boxed {4 x^{2} y^{\prime \prime }+8 y^{\prime } x -\left (4 x^{2}+12 x +3\right ) y=0} \]
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 &= \sqrt {x}\, {\mathrm e}^{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {4 x^{2} {\mathrm e}^{x} \operatorname {expIntegral}_{1}\left (2 x \right )-2 \,{\mathrm e}^{-x} x +{\mathrm e}^{-x}}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{2}} {\mathrm e}^{x} y}{4 x^{2} {\mathrm e}^{2 x} \operatorname {expIntegral}_{1}\left (2 x \right )-2 x +1}\right ] \\ \end{align*}