Link to actual problem [10940] \[ \boxed {x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (-1+2 b \right ) x^{2}+b \left (1+b \right )\right ) y=0} \]
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^{-b} {\mathrm e}^{-\frac {x^{2} a}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{b} {\mathrm e}^{\frac {x^{2} a}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-b} {\mathrm e}^{-\frac {x^{2} a}{2}} \left (\Gamma \left (b +\frac {1}{2}\right )-\Gamma \left (b +\frac {1}{2}, -x^{2} a \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{b} {\mathrm e}^{\frac {x^{2} a}{2}} y}{\Gamma \left (b +\frac {1}{2}\right )-\Gamma \left (b +\frac {1}{2}, -x^{2} a \right )}\right ] \\ \end{align*}