Link to actual problem [1126] \[ \boxed {x^{2} \ln \left (x \right )^{2} y^{\prime \prime }-2 x \ln \left (x \right ) y^{\prime }+\left (2+\ln \left (x \right )\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \ln \left (x \right ) \end {align*}
type detected by program
{"reduction_of_order", "second_order_change_of_variable_on_y_method_1", "second_order_ode_non_constant_coeff_transformation_on_B"}
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 [R &= x, S \left (R \right ) &= \frac {y}{\ln \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \ln \left (x \right )}\right ] \\ \end{align*}