Link to actual problem [9672] \[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}}=0} \]
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 \,{\mathrm e}^{\frac {1}{2 x^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{2 x^{2}}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \,{\mathrm e}^{\frac {1}{2 x^{2}}} \operatorname {erf}\left (\frac {\sqrt {2}}{2 x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{2 x^{2}}} y}{x \,\operatorname {erf}\left (\frac {\sqrt {2}}{2 x}\right )}\right ] \\ \end{align*}