Link to actual problem [9835] \[ \boxed {x^{3} y^{\prime \prime \prime }+3 y^{\prime \prime } x^{2}-2 y^{\prime } x +2 y=6 x^{3} \left (x -1\right ) \ln \left (x \right )-x^{3} \left (x +8\right )} \]
type detected by program
{"higher_order_ODE_non_constant_coefficients_of_type_Euler"}
type detected by Maple
[[_3rd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x}\right ] \\ \end{align*}
\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 &= x, S \left (R \right ) &= \frac {y}{x \ln \left (x \right )}\right ] \\ \end{align*}