Link to actual problem [10887] \[ \boxed {x y^{\prime \prime }+y^{\prime } a +\left (b x +c \right ) y=0} \]
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 &= {\mathrm e}^{-i \sqrt {b}\, x} \operatorname {KummerM}\left (\frac {a}{2}+\frac {i c}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{i \sqrt {b}\, x} y}{\operatorname {KummerM}\left (\frac {i c +a \sqrt {b}}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-i \sqrt {b}\, x} \operatorname {KummerU}\left (\frac {a}{2}+\frac {i c}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{i \sqrt {b}\, x} y}{\operatorname {KummerU}\left (\frac {i c +a \sqrt {b}}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right )}\right ] \\ \end{align*}