Link to actual problem [3853] \[ \boxed {\left (1-x^{2}+y^{2}\right ) y^{\prime }+y^{2}=x^{2}+1} \]
type detected by program
{"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 {x -y}{x^{2}-y^{2}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {y^{2}}{2}-x y-\ln \left (y-x \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 {2 x -2 y}{x^{2}-y^{2}-1} \\ \frac {dS}{dR} &= -\frac {R}{2} \\ \end{align*}