Link to actual problem [15051] \[ \boxed {\left (x^{3}+{\mathrm e}^{y}\right ) y^{\prime }=3 x^{2}} \]
type detected by program
{"exactWithIntegrationFactor", "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 &= \frac {x}{3}, \underline {\hspace {1.25 ex}}\eta &= 1\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x \left (y +1\right )}{3}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {{\mathrm e}^{y} y}{x^{3}}, S \left (R \right ) &= \ln \left (y\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 {3 \,{\mathrm e}^{y}}{x^{3}+{\mathrm e}^{y}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}