Link to actual problem [11035] \[ \boxed {x^{4} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y=0} \]
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 \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {c}}, \frac {\sqrt {1-4 a}}{2}, \frac {2 i \sqrt {c}}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {c}}, \frac {\sqrt {1-4 a}}{2}, \frac {2 i \sqrt {c}}{x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {c}}, \frac {\sqrt {1-4 a}}{2}, \frac {2 i \sqrt {c}}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {c}}, \frac {\sqrt {1-4 a}}{2}, \frac {2 i \sqrt {c}}{x}\right )}\right ] \\ \end{align*}