Link to actual problem [8993] \[ \boxed {y^{\prime }-x \sqrt {a^{2} x^{2}+2 a b x +4 y a +b^{2}-4 c}=-\frac {x a}{2}-\frac {b}{2}} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(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 {2 x}{a}, \underline {\hspace {1.25 ex}}\eta &= \frac {a^{2} x^{2}+3 a b x +8 a y +2 b^{2}-8 c}{a^{2}}\right ] \\ \left [R &= \frac {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c}{4 x^{4} a}, S \left (R \right ) &= \frac {a \ln \left (x \right )}{2}\right ] \\ \end{align*}
My program’s symgen result This shows my program’s found \(\xi ,\eta \) and the corresponding ODE in canonical coordinates \(R,S\).\begin{align*} \xi &= 0 \\ \eta &=\frac {a^{2} x^{2}-\sqrt {a^{2} x^{2}+2 a b x +4 a y +b^{2}-4 c}\, a \,x^{2}+2 a b x +4 a y +b^{2}-4 c}{a} \\ \frac {dS}{dR} &= 0 \\ \end{align*}