Link to actual problem [8220] \[ \boxed {x \left (x -1\right )^{2} y^{\prime \prime }-2 y=0} \]
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 {x}{-1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right ) y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {2 x \ln \left (x \right )-x^{2}+1}{-1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right ) y}{2 x \ln \left (x \right )-x^{2}+1}\right ] \\ \end{align*}