Link to actual problem [8058] \[ \boxed {3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y=0} \]
type detected by program
{"kovacic"}
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^{\frac {1}{3}}}{\left (x^{2}+3\right )^{\frac {2}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+3\right )^{\frac {2}{3}} y}{x^{\frac {1}{3}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{\frac {1}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}} \sqrt {3}}{6+\left (9 x^{2}+27\right )^{\frac {1}{3}}}\right )-\ln \left (1+\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}} \left (\left (9 x^{2}+27\right )^{\frac {1}{3}}+3\right )}{9}\right )+2 \ln \left (1-\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}}}{3}\right )\right )}{\left (9 x^{2}+27\right )^{\frac {2}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (9 x^{2}+27\right )^{\frac {2}{3}} y}{x^{\frac {1}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}} \sqrt {3}}{6+\left (9 x^{2}+27\right )^{\frac {1}{3}}}\right )-\ln \left (1+\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}}}{3}+\frac {\left (9 x^{2}+27\right )^{\frac {2}{3}}}{9}\right )+2 \ln \left (1-\frac {\left (9 x^{2}+27\right )^{\frac {1}{3}}}{3}\right )\right )}\right ] \\ \end{align*}