Link to actual problem [9676] \[ \boxed {y^{\prime \prime }+\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}}=0} \]
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 {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^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{4 x^{2}}} x^{2} 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*}
\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 (1, \frac {1}{4 x^{2}}\right )+\operatorname {BesselI}\left (0, \frac {1}{4 x^{2}}\right )\right )}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{4 x^{2}}} x^{2} 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*}