Link to actual problem [1107] \[ \boxed {\left (1+2 x \right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y=\left (1+2 x \right )^{2}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x} \end {align*}
type detected by program
{"reduction_of_order"}
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 {{\mathrm e}^{-x} y}{x}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}