Link to actual problem [14772] \[ \boxed {x^{4} y^{\prime \prime \prime \prime }+8 x^{3} y^{\prime \prime \prime }+27 x^{2} y^{\prime \prime }+35 x y^{\prime }+45 y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {y}{\sin \left (3 \ln \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\cos \left (3 \ln \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sin \left (2 \ln \left (x \right )\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\sin \left (2 \ln \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\cos \left (2 \ln \left (x \right )\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\cos \left (2 \ln \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}