Link to actual problem [11880] \[ \boxed {x^{2} y^{\prime \prime }-6 y=\ln \left (x \right )} \] With initial conditions \begin {align*} \left [y \left (1\right ) = {\frac {1}{6}}, y^{\prime }\left (1\right ) = -{\frac {1}{6}}\right ] \end {align*}
type detected by program
{"kovacic", "second_order_euler_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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{x^{2}}\right ] \\ \\ \end{align*}
\begin{align*} \\ \left [R &= y+\frac {\ln \left (x \right )}{6}, S \left (R \right ) &= -\frac {\ln \left (x \right )}{6}\right ] \\ \end{align*}