Link to actual problem [5889] \[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}}=0} \]
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 {{\mathrm e}^{\sqrt {x}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\sqrt {x}} x y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\sqrt {x}} x^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\sqrt {x}} y}{x^{2}}\right ] \\ \end{align*}