Link to actual problem [3888] \[ \boxed {x \left (1-x^{2}+y^{2}\right ) y^{\prime }+\left (1+x^{2}-y^{2}\right ) y=0} \]
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}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {-y-\frac {x^{2}-1}{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}-1\right )}{x^{2}-y^{2}-1}\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}-2 y^{2}+1}{x \left (x^{2}-y^{2}-1\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\sqrt {2}\, \ln \left (2 \,\operatorname {csgn}\left (x \right ) x y-\sqrt {2}\, y^{2}-\sqrt {2}\, x^{2}+\sqrt {2}\right )}{4 \,\operatorname {csgn}\left (x \right )}+\frac {\sqrt {2}\, \ln \left (\sqrt {2}\, y^{2}+\sqrt {2}\, x^{2}+2 \,\operatorname {csgn}\left (x \right ) x y-\sqrt {2}\right )}{4 \,\operatorname {csgn}\left (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 {-x^{2} y -y^{3}+y}{x^{2}-y^{2}-1} \\ \frac {dS}{dR} &= \frac {1}{R} \\ \end{align*}