Link to actual problem [9729] \[ \boxed {y^{\prime \prime }+\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}+\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (1+a \right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}}=0} \]
type detected by program
{"unknown"}
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 &= -\left (x^{2}-1\right )^{a} \operatorname {HeunC}\left (0, v , 1, \frac {1}{4}, \frac {a}{2}+\frac {1}{4}, x^{2}\right ) \left (x^{2+v}-x^{v}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-v} \left (x^{2}-1\right )^{-a} y}{\left (-x^{2}+1\right ) \operatorname {HeunC}\left (0, v , 1, \frac {1}{4}, \frac {a}{2}+\frac {1}{4}, x^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\left (x^{2}-1\right )^{a} \operatorname {HeunC}\left (0, -v , 1, \frac {1}{4}, \frac {a}{2}+\frac {1}{4}, x^{2}\right ) \left (x^{-v +2}-x^{-v}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-1\right )^{-a} x^{v} y}{\left (-x^{2}+1\right ) \operatorname {HeunC}\left (0, -v , 1, \frac {1}{4}, \frac {a}{2}+\frac {1}{4}, x^{2}\right )}\right ] \\ \end{align*}