Link to actual problem [8002] \[ \boxed {2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (1-2 x \right ) y=0} \]
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}\, \sinh \left (2 \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \sinh \left (2 \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \cosh \left (2 \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \cosh \left (2 \sqrt {x}\right )}\right ] \\ \end{align*}