Link to actual problem [9675] \[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}=0} \]
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 &= \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{\left (-1+x \right ) \left (1+x \right )}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{\left (-1+x \right ) \left (1+x \right )}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{\left (-1+x \right ) \left (1+x \right )}\right ) \left (\int \frac {1}{x \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{\left (-1+x \right ) \left (1+x \right )}\right )^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{\left (-1+x \right ) \left (1+x \right )}\right ) \left (\int \frac {1}{x \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{\left (-1+x \right ) \left (1+x \right )}\right )^{2}}d x \right )}\right ] \\ \end{align*}