Link to actual problem [3891] \[ \boxed {\left (x \left (a -x^{2}-y^{2}\right )+y\right ) y^{\prime }-\left (a -x^{2}-y^{2}\right ) y=-x} \]
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 {\left (x^{2}+y^{2}\right ) \left (-x^{2}-y^{2}+a \right )}{-x^{3}-x \,y^{2}+x a +y}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (x^{2}+y^{2}\right )}{2 a}-\arctan \left (\frac {y}{x}\right )+\frac {\ln \left (y^{2}+x^{2}-a \right )}{2 a}\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^{4}-2 x^{2} y^{2}-y^{4}+x^{2} a +a \,y^{2}}{-x^{3}-x \,y^{2}+x a +y} \\ \frac {dS}{dR} &= 0 \\ \end{align*}