Link to actual problem [6179] \[ \boxed {y^{\prime } x -x^{3} \left (y-1\right ) y^{\prime }=-2} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]
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 &= -\frac {x^{3}}{2}, \underline {\hspace {1.25 ex}}\eta &= 1\right ] \\ \left [R &= y-\frac {1}{x^{2}}, S \left (R \right ) &= \frac {1}{x^{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 -1}{x^{2} y -x^{2}-1} \\ \frac {dS}{dR} &= -\frac {2}{R} \\ \end{align*}