Link to actual problem [12783] \[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y=3 x +4} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1, y^{\prime \prime \prime }\left (0\right ) = 1] \end {align*}
type detected by program
{"higher_order_linear_constant_coefficients_ODE"}
type detected by Maple
[[_high_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}{\sin \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\cos \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \sin \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\cos \left (x \right ) x}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {9 x}{4}+\frac {3 y}{4}\right ] \\ \left [R &= -\left (4+3 x -y\right ) {\mathrm e}^{-\frac {3 x}{4}}, S \left (R \right ) &= x\right ] \\ \end{align*}