Link to actual problem [9890] \[ \boxed {x^{4} y^{\prime \prime \prime \prime }+8 x^{3} y^{\prime \prime \prime }+12 y^{\prime \prime } x^{2}+y a=0} \]
type detected by program
{"higher_order_ODE_non_constant_coefficients_of_type_Euler"}
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {1}{2}-\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x}\, x^{\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {1}{2}+\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x}\, x^{-\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {1}{2}-\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x}\, x^{\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {1}{2}+\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {x}\, x^{-\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}} y\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}