Link to actual problem [9617] \[ \boxed {16 x^{2} y^{\prime \prime }+32 y^{\prime } x -\left (4 x +5\right ) y=0} \]
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 &= \left (\frac {1}{x^{\frac {3}{4}}}-\frac {1}{x^{\frac {5}{4}}}\right ) {\mathrm e}^{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\sqrt {x}} x^{\frac {5}{4}} y}{-1+\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\sqrt {x}} \left (\sqrt {x}+1\right )}{x^{\frac {5}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\sqrt {x}} x^{\frac {5}{4}} y}{\sqrt {x}+1}\right ] \\ \end{align*}