Link to actual problem [15396] \[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+y=x \left (6-\ln \left (x \right )\right )} \]
type detected by program
{"kovacic", "second_order_euler_ode", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2", "second_order_change_of_variable_on_y_method_2"}
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 &= {\mathrm e}^{i \sqrt {\frac {1}{x^{2}}}\, x \ln \left (x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{-i \sqrt {\frac {1}{x^{2}}}\, x} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x, \underline {\hspace {1.25 ex}}\eta &= y -\frac {x}{2}\right ] \\ \left [R &= \frac {x \ln \left (x \right )+2 y}{2 x}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}