Link to actual problem [7624] \[ \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (1-x \right ) y=0} \]
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^{2}+2 x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x^{2}+2 x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {x}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {2 x +4}}{2}\right ) x -\sqrt {2}\, \sqrt {2+x}+2 \,\operatorname {arctanh}\left (\frac {\sqrt {2 x +4}}{2}\right )\right )}{\sqrt {2+x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {2+x}\, y}{\left (\left (2+x \right ) \operatorname {arctanh}\left (\frac {\sqrt {2 x +4}}{2}\right )-\sqrt {2}\, \sqrt {2+x}\right ) \sqrt {x}}\right ] \\ \end{align*}