Link to actual problem [10829] \[ \boxed {y^{\prime \prime }-\left (a \,x^{2}+b c x \right ) y=0} \]
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 {hypergeom}\left (\left [-\frac {b^{2} c^{2}}{16 a^{\frac {3}{2}}}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\left (2 x a +b c \right )^{2}}{4 a^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {x \left (x a +b c \right )}{2 \sqrt {a}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (x a +b c \right )}{2 \sqrt {a}}} y}{\operatorname {hypergeom}\left (\left [-\frac {b^{2} c^{2}-4 a^{\frac {3}{2}}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (2 x a +b c \right )^{2}}{4 a^{\frac {3}{2}}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {b^{2} c^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (2 x a +b c \right )^{2}}{4 a^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {x \left (x a +b c \right )}{2 \sqrt {a}}} \left (2 x a +b c \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (x a +b c \right )}{2 \sqrt {a}}} y}{\operatorname {hypergeom}\left (\left [-\frac {b^{2} c^{2}-12 a^{\frac {3}{2}}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (2 x a +b c \right )^{2}}{4 a^{\frac {3}{2}}}\right ) \left (2 x a +b c \right )}\right ] \\ \end{align*}