Link to actual problem [12279] \[ \boxed {\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime \prime }+\left (7 \sin \left (x \right )+4 \cos \left (x \right )\right ) y^{\prime }+10 \cos \left (x \right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_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 &= {\mathrm e}^{-2 \ln \left (2 \tan \left (x \right )-1\right )+\frac {\ln \left (\tan \left (x \right )^{2}+1\right )}{2}+\ln \left (\tan \left (x \right )\right )-2 x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 \tan \left (x \right )-1\right )^{2} {\mathrm e}^{2 x} y}{\sqrt {\tan \left (x \right )^{2}+1}\, \tan \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-2 \ln \left (2 \tan \left (x \right )-1\right )+\frac {\ln \left (\tan \left (x \right )^{2}+1\right )}{2}+\ln \left (\tan \left (x \right )\right )-2 x} \left (\int -\frac {\csc \left (x \right ) {\mathrm e}^{2 \ln \left (2 \tan \left (x \right )-1\right )-\frac {\ln \left (\tan \left (x \right )^{2}+1\right )}{2}-\ln \left (\tan \left (x \right )\right )+2 x}}{-2 \sin \left (x \right )+\cos \left (x \right )}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 \tan \left (x \right )-1\right )^{2} {\mathrm e}^{2 x} y}{\sqrt {\tan \left (x \right )^{2}+1}\, \tan \left (x \right ) \left (\int -\frac {\csc \left (x \right ) {\mathrm e}^{2 x} \left (2 \tan \left (x \right )-1\right )^{2}}{\sqrt {\tan \left (x \right )^{2}+1}\, \tan \left (x \right ) \left (-2 \sin \left (x \right )+\cos \left (x \right )\right )}d x \right )}\right ] \\ \end{align*}