Link to actual problem [15416] \[ \boxed {x \left (x -1\right ) y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+2 y=\left (2 x -3\right ) x^{2}} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
type detected by program
{"reduction_of_order", "second_order_change_of_variable_on_y_method_2", "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*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {1}{2}+x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{-\frac {1}{2}+x}\right ] \\ \end{align*}