Link to actual problem [8728] \[ \boxed {{y^{\prime }}^{2}-y x y^{\prime }+y^{2} \ln \left (a y\right )=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
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 y}{2}\right ] \\ \left [R &= y \,{\mathrm e}^{-\frac {x^{2}}{4}}, S \left (R \right ) &= x\right ] \\ \end{align*}