Link to actual problem [4139] \[ \boxed {x^{2} {y^{\prime }}^{2}+2 x \left (2 x +y\right ) y^{\prime }+y^{2}=4 a} \]
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 [R &= \frac {\sqrt {-a \left (-y^{2}+4 a \right )}\, x}{2 a \left (x y+2 a \right )}, S \left (R \right ) &= -\frac {\operatorname {arctanh}\left (\frac {y}{2 \sqrt {a}}\right )}{2 \sqrt {a}}\right ] \\ \end{align*}
\begin{align*} \\ \text {Expression too large to display} \\ \end{align*}