Link to actual problem [10145] \[ \boxed {\left ({y^{\prime }}^{2}+a \left (y^{\prime } x -y\right )\right ) y^{\prime \prime }=b} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {x a}{2}\right ] \\ \left [R &= y+\frac {x^{2} a}{4}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (\textit {\_y1} \,a^{2} x +\textit {\_y1}^{2} a -a^{2} y+2 b \right )}{a^{2}}\right ] \\ \end{align*}