Link to actual problem [9633] \[ \boxed {x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (x^{2} a +b x +a \right ) y=0} \]
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 {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {1+x}{-1+x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {1+x}{-1+x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {1+x}{-1+x}\right ) \left (\int \frac {1}{x \operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {1+x}{-1+x}\right )^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {1+x}{-1+x}\right ) \left (\int \frac {1}{x \operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {1+x}{-1+x}\right )^{2}}d x \right )}\right ] \\ \end{align*}