Link to actual problem [3204] \[ \boxed {y^{2}+\left (y x +y^{2}-1\right ) y^{\prime }=0} \]
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}{x y +y^{2}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y^{2}}{2}+x y-\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 {y}{x y +y^{2}-1} \\ \frac {dS}{dR} &= 0 \\ \end{align*}