Link to actual problem [7208] \[ \boxed {x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y x=0} \]
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 &= x^{-\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}} \left (-47+3 \sqrt {249}\right )}{768}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \ln \left (x \right ) \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}+\frac {2 \sqrt {3}\, \ln \left (x \right )}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}} \left (-47+3 \sqrt {249}\right ) \left (-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}\right )}{1536}} y}{\sin \left (\frac {\sqrt {3}\, \ln \left (x \right ) \left (3 \left (188+12 \sqrt {249}\right )^{{2}/{3}} \sqrt {249}+16 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-47 \left (188+12 \sqrt {249}\right )^{{2}/{3}}\right )}{192}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \ln \left (x \right ) \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}+\frac {2 \sqrt {3}\, \ln \left (x \right )}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}} \left (-47+3 \sqrt {249}\right ) \left (-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}\right )}{1536}} y}{\cos \left (\frac {\sqrt {3}\, \ln \left (x \right ) \left (3 \left (188+12 \sqrt {249}\right )^{{2}/{3}} \sqrt {249}+16 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-47 \left (188+12 \sqrt {249}\right )^{{2}/{3}}\right )}{192}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}