Link to actual problem [14769] \[ \boxed {x^{3} y^{\prime \prime \prime }+16 x^{2} y^{\prime \prime }+79 x y^{\prime }+125 y=0} \]
type detected by program
{"higher_order_ODE_non_constant_coefficients_of_type_Euler"}
type detected by Maple
[[_3rd_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 &= \frac {1}{x^{5}}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sin \left (3 \ln \left (x \right )\right )}{x^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{4} y}{\sin \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 {\cos \left (3 \ln \left (x \right )\right )}{x^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{4} y}{\cos \left (3 \ln \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}