Link to actual problem [8994] \[ \boxed {y^{\prime }-x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}=-\frac {x}{2}-\frac {a}{2}} \]
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 &= x, \underline {\hspace {1.25 ex}}\eta &= \frac {3}{2} a^{2}+\frac {5}{2} x a +x^{2}+6 y\right ] \\ \left [R &= \frac {x^{2}+2 x a +a^{2}+4 y}{4 x^{6}}, S \left (R \right ) &= \ln \left (x \right )\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 &=-x^{3} \sqrt {a^{2}+2 x a +x^{2}+4 y}+\frac {3 a^{2}}{2}+3 x a +\frac {3 x^{2}}{2}+6 y \\ \frac {dS}{dR} &= 0 \\ \end{align*}