Link to actual problem [3889] \[ \boxed {x \left (a -x^{2}-y^{2}\right ) y^{\prime }+\left (a +x^{2}+y^{2}\right ) y=0} \]
type detected by program
{"exactWithIntegrationFactor", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
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^{2}}{-x^{2}-y^{2}+a}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {-y-\frac {-x^{2}+a}{y}}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (-x^{2}+y^{2}+a \right )}{-x^{2}-y^{2}+a}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{4}+y^{4}-2 x^{2} a +2 a \,y^{2}+a^{2}}{x \left (-x^{2}-y^{2}+a \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\sqrt {2}\, \arctan \left (\frac {\left (2 y+\sqrt {2 x^{2}-4 a}\right ) \sqrt {2}}{2 x}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2 y+\sqrt {2 x^{2}-4 a}\right ) \sqrt {2}}{2 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 {-a \,x^{2} y +a \,y^{3}+a^{2} y}{-x^{2}-y^{2}+a} \\ \frac {dS}{dR} &= -\frac {1}{R a} \\ \end{align*}