Link to actual problem [9632] \[ \boxed {x^{3} y^{\prime \prime }+2 y^{\prime } x -y=0} \]
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 &= {\mathrm e}^{\frac {1}{x}} \left (\operatorname {BesselI}\left (0, \frac {1}{x}\right )-\operatorname {BesselI}\left (1, \frac {1}{x}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{x}} y}{\operatorname {BesselI}\left (0, \frac {1}{x}\right )-\operatorname {BesselI}\left (1, \frac {1}{x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {1}{x}} \left (\operatorname {BesselK}\left (0, -\frac {1}{x}\right )-\operatorname {BesselK}\left (1, -\frac {1}{x}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{x}} y}{\operatorname {BesselK}\left (0, -\frac {1}{x}\right )-\operatorname {BesselK}\left (1, -\frac {1}{x}\right )}\right ] \\ \end{align*}