Link to actual problem [8034] \[ \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (x +1\right ) y=0} \]
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 {x}}{\left (2+x \right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2+x \right )^{\frac {3}{2}} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {x}\, \sqrt {2}}{2+x}-\frac {2 \sqrt {x}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {2+x}}{2}\right )}{\left (2+x \right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2+x \right )^{\frac {5}{2}} y}{\sqrt {x}\, \left (\sqrt {2}\, \left (2+x \right )^{\frac {3}{2}}-2 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {2+x}}{2}\right ) x -4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {2+x}}{2}\right )\right )}\right ] \\ \end{align*}