Link to actual problem [9900] \[ \boxed {y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y-\lambda \left (a x -b \right ) \left (y^{\prime \prime }-a^{2} y\right )=0} \]
type detected by program
{"unknown"}
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {{\mathrm e}^{x a} {\mathrm e}^{-2 x a}}{2 a}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{x a} \left (\int {\mathrm e}^{-2 x a} \left (\int {\mathrm e}^{x a} \operatorname {AiryAi}\left (-\frac {\left (-a \lambda \right )^{{1}/{3}} \left (a \lambda x +a^{2}-b \lambda \right )}{a \lambda }\right )d x \right )d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x a} y}{\int {\mathrm e}^{-2 x a} \left (\int {\mathrm e}^{x a} \operatorname {AiryAi}\left (-\frac {\left (-a \lambda \right )^{{1}/{3}} \left (\lambda \left (x a -b \right )+a^{2}\right )}{a \lambda }\right )d x \right )d x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{x a} \left (\int {\mathrm e}^{-2 x a} \left (\int {\mathrm e}^{x a} \operatorname {AiryBi}\left (-\frac {\left (-a \lambda \right )^{{1}/{3}} \left (a \lambda x +a^{2}-b \lambda \right )}{a \lambda }\right )d x \right )d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x a} y}{\int {\mathrm e}^{-2 x a} \left (\int {\mathrm e}^{x a} \operatorname {AiryBi}\left (-\frac {\left (-a \lambda \right )^{{1}/{3}} \left (\lambda \left (x a -b \right )+a^{2}\right )}{a \lambda }\right )d x \right )d x}\right ] \\ \end{align*}