Link to actual problem [15085] \[ \boxed {-y x +\left (x^{2}+y\right ) y^{\prime }=-x} \]
type detected by program
{"exactWithIntegrationFactor", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class A`]]
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 (y -1\right )^{3}}{x^{2}+y}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {1}{y-1}-\frac {x^{2}+1}{2 \left (y-1\right )^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (y -1\right ) \left (x^{2}+2 y -1\right )}{x^{2}+y}\right ] \\ \left [R &= x, S \left (R \right ) &= \ln \left (y-1\right )-\frac {\ln \left (x^{2}+2 y-1\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 +y^{3}-x^{2}-y^{2}}{x^{2}+y} \\ \frac {dS}{dR} &= 0 \\ \end{align*}