Link to actual problem [9636] \[ \boxed {x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+x y=0} \]
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 &= \frac {{\mathrm e}^{\frac {1}{4 x^{2}}} \left (2 \operatorname {BesselI}\left (0, \frac {1}{4 x^{2}}\right ) x^{2}-\operatorname {BesselI}\left (0, \frac {1}{4 x^{2}}\right )+\operatorname {BesselI}\left (1, \frac {1}{4 x^{2}}\right )\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{4 x^{2}}} x y}{2 \operatorname {BesselI}\left (0, \frac {1}{4 x^{2}}\right ) x^{2}-\operatorname {BesselI}\left (0, \frac {1}{4 x^{2}}\right )+\operatorname {BesselI}\left (1, \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 &= \frac {{\mathrm e}^{\frac {1}{4 x^{2}}} \left (2 \operatorname {BesselK}\left (0, -\frac {1}{4 x^{2}}\right ) x^{2}-\operatorname {BesselK}\left (0, -\frac {1}{4 x^{2}}\right )+\operatorname {BesselK}\left (1, -\frac {1}{4 x^{2}}\right )\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{4 x^{2}}} x y}{2 \operatorname {BesselK}\left (0, -\frac {1}{4 x^{2}}\right ) x^{2}-\operatorname {BesselK}\left (0, -\frac {1}{4 x^{2}}\right )+\operatorname {BesselK}\left (1, -\frac {1}{4 x^{2}}\right )}\right ] \\ \end{align*}