Link to actual problem [7892] \[ \boxed {\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y=0} \]
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 &= x^{4}-9 x^{2}+\frac {27}{8}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{4}-9 x^{2}+\frac {27}{8}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (4 x^{4}-36 x^{2}+\frac {27}{2}\right ) \ln \left (\sqrt {x^{2}+3}-x \right )+\frac {225}{8}+\frac {25 \sqrt {x^{2}+3}\, x^{3}}{3}+\frac {25 x^{4}}{3}-\frac {55 \sqrt {x^{2}+3}\, x}{2}-75 x^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\frac {225}{8}+\frac {\left (8 x^{4}-72 x^{2}+27\right ) \ln \left (\sqrt {x^{2}+3}-x \right )}{2}+\frac {5 \left (10 x^{3}-33 x \right ) \sqrt {x^{2}+3}}{6}+\frac {25 x^{4}}{3}-75 x^{2}}\right ] \\ \end{align*}