Link to actual problem [9840] \[ \boxed {x^{3} y^{\prime \prime \prime }+\left (x +3\right ) x^{2} y^{\prime \prime }+5 \left (x -6\right ) x y^{\prime }+\left (4 x +30\right ) y=0} \]
type detected by program
{"unknown"}
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 {x^{4}-84 x^{3}+2016 x^{2}-20160 x +75600}{x^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{6} y}{x^{4}-84 x^{3}+2016 x^{2}-20160 x +75600}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-x} \left (x^{8}+28 x^{7}+450 x^{6}+5100 x^{5}+42900 x^{4}+267120 x^{3}+1179360 x^{2}+3326400 x +4536000\right )}{x^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} x^{6} y}{x^{8}+28 x^{7}+450 x^{6}+5100 x^{5}+42900 x^{4}+267120 x^{3}+1179360 x^{2}+3326400 x +4536000}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 29+\frac {{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right ) x^{8}+28 \,{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right ) x^{7}+450 \,{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right ) x^{6}+5100 \,{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right ) x^{5}+x^{7}+42900 \,{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right ) x^{4}+60 x^{4} \ln \left (x \right )+267120 \,{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right ) x^{3}+480 x^{5}-5040 x^{3} \ln \left (x \right )+1179360 x^{2} {\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right )+5612 x^{4}+120960 x^{2} \ln \left (x \right )+3326400 \,{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right ) x +40152 x^{3}-1209600 x \ln \left (x \right )+4536000 \,{\mathrm e}^{-x} \operatorname {Ei}_{1}\left (-x \right )+654192 x^{2}+4536000 \ln \left (x \right )-2761920 x +27367200}{x^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{6} {\mathrm e}^{x} y}{\left (x^{8}+28 x^{7}+450 x^{6}+5100 x^{5}+42900 x^{4}+267120 x^{3}+1179360 x^{2}+3326400 x +4536000\right ) \operatorname {Ei}_{1}\left (-x \right )+{\mathrm e}^{x} \left (27367200+60 \left (x^{4}-84 x^{3}+2016 x^{2}-20160 x +75600\right ) \ln \left (x \right )+x^{7}+29 x^{6}+480 x^{5}+5612 x^{4}+40152 x^{3}+654192 x^{2}-2761920 x \right )}\right ] \\ \end{align*}