Link to actual problem [10955] \[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } a x +\left (b \,x^{n}+c \right ) y=0} \]
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 &= x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a}{2}} y}{\sqrt {x}\, \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\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 {\sqrt {a^{2}-2 a -4 c +1}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a}{2}} y}{\sqrt {x}\, \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )}\right ] \\ \end{align*}