Link to actual problem [8329] \[ \boxed {y^{\prime \prime }-\left (\frac {1}{x}-\frac {3}{16 x^{2}}\right ) y=0} \]
type detected by program
{"kovacic", "second_order_bessel_ode"}
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 {1}{4}} \sinh \left (2 \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {1}{4}} \sinh \left (2 \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{4}} \cosh \left (2 \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {1}{4}} \cosh \left (2 \sqrt {x}\right )}\right ] \\ \end{align*}