Link to actual problem [1045] \[ \boxed {{\mathrm e}^{y x} \left (y x^{4}+4 x^{3}\right )+3 y+\left (x^{5} {\mathrm e}^{y x}+3 x \right ) y^{\prime }=0} \]
type detected by program
{"exact", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_exact, [_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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{x \left ({\mathrm e}^{x y} x^{4}+3\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 &=\frac {4 \,{\mathrm e}^{x y} x^{4}+12 x y}{{\mathrm e}^{x y} x^{5}+3 x} \\ \frac {dS}{dR} &= 0 \\ \end{align*}