Link to actual problem [8245] \[ \boxed {2 x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (2 x -1\right ) y=0} \]
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 &= \frac {\sqrt {\frac {8 x^{\frac {3}{2}}+4 i x +i+2 \sqrt {x}}{-2 \sqrt {x}+i}}\, {\mathrm e}^{2 i \sqrt {x}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{-2 i \sqrt {x}} y}{\sqrt {\frac {\left (2 \sqrt {x}+i\right ) \left (1+4 x \right )}{-2 \sqrt {x}+i}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {\frac {-8 x^{\frac {3}{2}}+4 i x +i-2 \sqrt {x}}{2 \sqrt {x}+i}}\, {\mathrm e}^{-2 i \sqrt {x}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{2 i \sqrt {x}} y}{\sqrt {\frac {\left (-2 \sqrt {x}+i\right ) \left (1+4 x \right )}{2 \sqrt {x}+i}}}\right ] \\ \end{align*}