Link to actual problem [9670] \[ \boxed {y^{\prime \prime }+\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}}=0} \]
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 &= 2 a x \operatorname {BesselI}\left (a , \frac {b}{x}\right )+\operatorname {BesselI}\left (a , \frac {b}{x}\right ) b +\operatorname {BesselI}\left (1+a , \frac {b}{x}\right ) b\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{2 a x \operatorname {BesselI}\left (a , \frac {b}{x}\right )+\operatorname {BesselI}\left (a , \frac {b}{x}\right ) b +\operatorname {BesselI}\left (1+a , \frac {b}{x}\right ) b}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 2 a x \operatorname {BesselK}\left (a , \frac {b}{x}\right )+\operatorname {BesselK}\left (a , \frac {b}{x}\right ) b -\operatorname {BesselK}\left (1+a , \frac {b}{x}\right ) b\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{2 a x \operatorname {BesselK}\left (a , \frac {b}{x}\right )+\operatorname {BesselK}\left (a , \frac {b}{x}\right ) b -\operatorname {BesselK}\left (1+a , \frac {b}{x}\right ) b}\right ] \\ \end{align*}