Link to actual problem [2819] \[ \boxed {x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+2 y=8 x^{2} {\mathrm e}^{2 x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
type detected by program
{"reduction_of_order", "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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x +\frac {1}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x +\frac {1}{2}}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}