Link to actual problem [9732] \[ \boxed {y^{\prime \prime }-\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {\left (x^{4} a +10 x^{2}+1\right ) y}{4 x^{6}}=0} \]
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^{\frac {3}{2}+\frac {\sqrt {-a +9}}{2}} {\mathrm e}^{-\frac {1}{4 x^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {\sqrt {-a +9}}{2}} {\mathrm e}^{\frac {1}{4 x^{2}}} y}{x^{\frac {3}{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {3}{2}-\frac {\sqrt {-a +9}}{2}} {\mathrm e}^{-\frac {1}{4 x^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {\sqrt {-a +9}}{2}} {\mathrm e}^{\frac {1}{4 x^{2}}} y}{x^{\frac {3}{2}}}\right ] \\ \end{align*}