Link to actual problem [8078] \[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y=0} \]
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^{3}}{\left (x^{2}+1\right )^{\frac {5}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{\frac {5}{2}} y}{x^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {3 x^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )}{\left (x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 x}{\left (x^{2}+1\right )^{2}}+\frac {2}{\left (x^{2}+1\right )^{2} x}+\frac {24}{\left (x^{2}+1\right )^{2} x^{3}}+\frac {16}{\left (x^{2}+1\right )^{2} x^{5}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{\frac {9}{2}} x^{5} y}{3 x^{8} \left (x^{2}+1\right )^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )-3 \left (x^{2}+2\right ) \left (x^{2}+1\right )^{\frac {5}{2}} \left (x^{4}-\frac {8}{3} x^{2}-\frac {8}{3}\right )}\right ] \\ \end{align*}