Link to actual problem [104] \[ \boxed {y^{3}+3 y^{2} y^{\prime }={\mathrm e}^{-x}} \]
type detected by program
{"bernoulli", "exactWithIntegrationFactor", "first_order_ode_lie_symmetry_lookup"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _Bernoulli]
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 &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {y}{3}\right ] \\ \left [R &= y \,{\mathrm e}^{\frac {x}{3}}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y \,{\mathrm e}^{\frac {x}{3}}}{\left (1+x \right )^{\frac {1}{3}}}, S \left (R \right ) &= -\frac {\ln \left (-x -1\right )}{3}\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 {{\mathrm e}^{-x}}{y^{2}} \\ \frac {dS}{dR} &= {\frac {1}{3}} \\ \end{align*}