Link to actual problem [8632] \[ \boxed {x \left (y^{2}+y x^{2}+x^{2}\right ) y^{\prime }-2 y^{3}-2 y^{2} x^{2}=-x^{4}} \]
type detected by program
{"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 {\left (x^{2}+2 y \right ) \left (x -y \right ) \left (x +y \right )}{x^{2} y +x^{2}+y^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (x^{2}+2 y\right )}{2}+\frac {\ln \left (y-x \right )}{2}+\frac {\ln \left (x +y\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^{5}-x^{3} y^{2}+2 x^{3} y -2 x \,y^{3}}{2 x^{3} y +2 x^{3}+2 x \,y^{2}} \\ \frac {dS}{dR} &= -\frac {2}{R} \\ \end{align*}