Link to actual problem [9663] \[ \boxed {y^{\prime \prime }+\frac {\left (-3 x +1\right ) y}{\left (x -1\right ) \left (2 x -1\right )^{2}}=0} \]
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 {2 x -1}\, \left (-1+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {2 x -1}\, \left (-1+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (2 \ln \left (-1+x \right ) x -2 \ln \left (2 x -1\right ) x -2 \ln \left (-1+x \right )+2 \ln \left (2 x -1\right )+1\right ) \sqrt {2 x -1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (\left (2 x -2\right ) \ln \left (-1+x \right )+1+\left (2-2 x \right ) \ln \left (2 x -1\right )\right ) \sqrt {2 x -1}}\right ] \\ \end{align*}