Link to actual problem [8326] \[ \boxed {y^{\prime \prime }-\frac {\left (4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4\right ) y}{4 x^{4}}=0} \]
type detected by program
{"kovacic", "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 &= \frac {{\mathrm e}^{\frac {x^{3}-2 x^{2}-2}{2 x}} \left (x^{2}-1\right )}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{2}} {\mathrm e}^{-\frac {x^{3}-2 x^{2}-2}{2 x}} y}{x^{2}-1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {x^{2}}{2}-x -\frac {1}{x}} \left (\int \frac {x^{3} {\mathrm e}^{-x^{2}+2 x +\frac {2}{x}}}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}d x \right ) \left (-1+\sqrt {x}\right ) \left (\sqrt {x}+1\right ) \left (1+x \right )}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{2}}{2}} {\mathrm e}^{x} {\mathrm e}^{\frac {1}{x}} x^{\frac {3}{2}} y}{\left (\int \frac {x^{3} {\mathrm e}^{-x^{2}} {\mathrm e}^{2 x} {\mathrm e}^{\frac {2}{x}}}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}d x \right ) \left (-1+\sqrt {x}\right ) \left (\sqrt {x}+1\right ) \left (1+x \right )}\right ] \\ \end{align*}