Link to actual problem [4156] \[ \boxed {\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 y y^{\prime } x=-x^{2}} \]
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 {-y^{2}+a^{2}-x^{2}}{x^{4}}, S \left (R \right ) &= \int _{}^{x}\frac {1}{\sqrt {-\frac {\left (-y^{2}+a^{2}-x^{2}\right ) \textit {\_a}^{4}}{x^{4}}-\textit {\_a}^{2}+a^{2}}\, \textit {\_a}}d \textit {\_a}\right ] \\ \end{align*}