Link to actual problem [3103] \[ \boxed {\left (x -1-y^{2}\right ) y^{\prime }-y=0} \]
type detected by program
{"exactWithIntegrationFactor", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational]
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 {y^{2}}{-y^{2}+x -1}\right ] \\ \left [R &= x, S \left (R \right ) &= -y-\frac {-1+x}{y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (y^{2}+x -1\right )}{-y^{2}+x -1}\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 {y^{2}}{-y^{2}+x -1} \\ \frac {dS}{dR} &= 0 \\ \end{align*}