Link to actual problem [4481] \[ \boxed {-y^{2}-y-\left (x^{2}-y^{2}-x \right ) y^{\prime }=-x^{2}} \]
type detected by program
{"exactByInspection", "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 {\left (x -y \right ) \left (x +y \right )}{x^{2}-y^{2}-x}\right ] \\ \left [R &= x, S \left (R \right ) &= -y+\frac {\ln \left (x +y\right )}{2}-\frac {\ln \left (y-x \right )}{2}\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 {-x^{2}+y^{2}}{x^{2}-y^{2}-x} \\ \frac {dS}{dR} &= -1 \\ \end{align*}