Link to actual problem [8849] \[ \boxed {f \left (y^{2}+x^{2}\right ) \left ({y^{\prime }}^{2}+1\right )-\left (y^{\prime } x -y\right )^{2}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= x^{2}+y^{2}, S \left (R \right ) &= -\arctan \left (\frac {x}{y}\right )\right ] \\ \end{align*}