Link to actual problem [9780] \[ \boxed {y^{\prime \prime \prime }+2 a x y^{\prime }+y a=0} \]
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 &= \operatorname {AiryAi}\left (-\frac {x 4^{{1}/{3}} a^{{1}/{3}}}{2}\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {AiryAi}\left (-\frac {x 4^{{1}/{3}} a^{{1}/{3}}}{2}\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {AiryBi}\left (-\frac {x 4^{{1}/{3}} a^{{1}/{3}}}{2}\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {AiryBi}\left (-\frac {x 4^{{1}/{3}} a^{{1}/{3}}}{2}\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {AiryAi}\left (-\frac {x 4^{{1}/{3}} a^{{1}/{3}}}{2}\right ) \operatorname {AiryBi}\left (-\frac {x 4^{{1}/{3}} a^{{1}/{3}}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {AiryAi}\left (-\frac {x 4^{{1}/{3}} a^{{1}/{3}}}{2}\right ) \operatorname {AiryBi}\left (-\frac {x 4^{{1}/{3}} a^{{1}/{3}}}{2}\right )}\right ] \\ \end{align*}