Link to actual problem [10948] \[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -\left (x^{2}+\left (n +\frac {1}{2}\right )^{2}\right ) y=0} \]
type detected by program
{"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 &= \operatorname {BesselI}\left (n +\frac {1}{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselI}\left (n +\frac {1}{2}, x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {BesselK}\left (n +\frac {1}{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselK}\left (n +\frac {1}{2}, x\right )}\right ] \\ \end{align*}