Link to actual problem [8950] \[ \boxed {y^{\prime }+\frac {-x^{2}+2 x^{3} y-F \left (\left (x y-1\right ) x \right )}{x^{4}}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_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 &= -\frac {x^{2}}{2}, \underline {\hspace {1.25 ex}}\eta &= x y -\frac {1}{2}\right ] \\ \left [R &= \left (x y-1\right ) x, S \left (R \right ) &= \frac {2}{x}\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 &= -\frac {x^{2}}{2} \\ \eta &=x y -\frac {1}{2} \\ \frac {dS}{dR} &= -\frac {2}{F \left (R \right )} \\ \end{align*}