Link to actual problem [9434] \[ \boxed {y^{\prime \prime } x +a y^{\prime }+y b x=0} \]
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 &= x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {a}{2}-\frac {1}{2}, x \sqrt {b}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a}{2}} y}{\sqrt {x}\, \operatorname {BesselJ}\left (\frac {a}{2}-\frac {1}{2}, x \sqrt {b}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselY}\left (\frac {a}{2}-\frac {1}{2}, x \sqrt {b}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a}{2}} y}{\sqrt {x}\, \operatorname {BesselY}\left (\frac {a}{2}-\frac {1}{2}, x \sqrt {b}\right )}\right ] \\ \end{align*}