Link to actual problem [10963] \[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+\left (b a -1\right ) x +b \right ) y^{\prime }+a^{2} b x y=0} \]
type detected by program
{"kovacic"}
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^{1-\frac {a b}{2}} \operatorname {HeunD}\left (-4 \sqrt {a b}, -4-\frac {b^{3} a^{3}+2 \left (a b \right )^{\frac {3}{2}} a b -2 \sqrt {a b}\, a^{2} b^{2}-4 a^{2} b^{2}+2 \left (a b \right )^{\frac {3}{2}}+6 \sqrt {a b}\, a b}{a b}, -8 \sqrt {a b}\, \left (a b -1\right ), 4-\frac {-b^{3} a^{3}+2 \left (a b \right )^{\frac {3}{2}} a b -2 \sqrt {a b}\, a^{2} b^{2}+4 a^{2} b^{2}+2 \left (a b \right )^{\frac {3}{2}}+6 \sqrt {a b}\, a b}{a b}, \frac {\sqrt {a b}\, x -b}{\sqrt {a b}\, x +b}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a b}{2}} y}{x \operatorname {HeunD}\left (-4 \sqrt {a b}, -a^{2} b^{2}+4 a b -8 \sqrt {a b}-4, -8 \sqrt {a b}\, \left (a b -1\right ), a^{2} b^{2}-4 a b -8 \sqrt {a b}+4, \frac {\sqrt {a b}\, x -b}{\sqrt {a b}\, x +b}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{1-\frac {a b}{2}} {\mathrm e}^{-\frac {x^{2} a -b}{x}} \operatorname {HeunD}\left (4 \sqrt {a b}, -4-\frac {b^{3} a^{3}+2 \left (a b \right )^{\frac {3}{2}} a b -2 \sqrt {a b}\, a^{2} b^{2}-4 a^{2} b^{2}+2 \left (a b \right )^{\frac {3}{2}}+6 \sqrt {a b}\, a b}{a b}, -8 \sqrt {a b}\, \left (a b -1\right ), 4-\frac {-b^{3} a^{3}+2 \left (a b \right )^{\frac {3}{2}} a b -2 \sqrt {a b}\, a^{2} b^{2}+4 a^{2} b^{2}+2 \left (a b \right )^{\frac {3}{2}}+6 \sqrt {a b}\, a b}{a b}, \frac {\sqrt {a b}\, x -b}{\sqrt {a b}\, x +b}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a b}{2}} {\mathrm e}^{\frac {x^{2} a -b}{x}} y}{x \operatorname {HeunD}\left (4 \sqrt {a b}, -a^{2} b^{2}+4 a b -8 \sqrt {a b}-4, -8 \sqrt {a b}\, \left (a b -1\right ), a^{2} b^{2}-4 a b -8 \sqrt {a b}+4, \frac {\sqrt {a b}\, x -b}{\sqrt {a b}\, x +b}\right )}\right ] \\ \end{align*}