Link to actual problem [9713] \[ \boxed {y^{\prime \prime }-\frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )}=0} \]
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^{2}+x +1}{\left (2 x +1\right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x +1\right )^{2} y}{x^{2}+x +1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (36 x^{2}+36 x +36\right ) \arctan \left (\frac {\sqrt {3}\, \left (2 x +1\right )}{3}\right )-16 \left (x^{3}+x^{2}+\frac {11}{8} x +\frac {3}{16}\right ) \sqrt {3}}{\left (2 x +1\right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x +1\right )^{2} y}{\left (36 x^{2}+36 x +36\right ) \arctan \left (\frac {\sqrt {3}\, \left (2 x +1\right )}{3}\right )-16 \left (x^{3}+x^{2}+\frac {11}{8} x +\frac {3}{16}\right ) \sqrt {3}}\right ] \\ \end{align*}