Link to actual problem [8401] \[ \boxed {y^{\prime }-\sqrt {\frac {a y^{2}+y b +c}{a \,x^{2}+b x +c}}=0} \]
type detected by program
{"exactWithIntegrationFactor", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
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 &= \frac {\left (x^{2} a +b x +c \right ) \left (2 a y +b \right )}{b^{2}}, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (a \,y^{2}+b y +c \right ) \left (2 x a +b \right )}{b^{2}}\right ] \\ \left [R &= \frac {a y^{2}+b y+c}{x^{2} a +b x +c}, S \left (R \right ) &= \int _{}^{x}\frac {b^{2}}{\left (\textit {\_a}^{2} a +\textit {\_a} b +c \right ) \sqrt {\frac {4 \left (a y^{2}+b y+c \right ) \textit {\_a}^{2} a^{2}}{x^{2} a +b x +c}+\frac {4 \left (a y^{2}+b y+c \right ) \textit {\_a} a b}{x^{2} a +b x +c}+\frac {4 \left (a y^{2}+b y+c \right ) a c}{x^{2} a +b x +c}-4 a c +b^{2}}}d \textit {\_a}\right ] \\ \end{align*}