Link to actual problem [8423] \[ \boxed {y^{\prime }-\frac {y-x f \left (x^{2}+a y^{2}\right )}{x +a y f \left (x^{2}+a y^{2}\right )}=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {a \,y^{2}+x^{2}}{x +a y f \left (a \,y^{2}+x^{2}\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {x +a \textit {\_a} f \left (\textit {\_a}^{2} a +x^{2}\right )}{\textit {\_a}^{2} a +x^{2}}d \textit {\_a}\right ] \\ \end{align*}