Link to actual problem [11608] \[ \boxed {\frac {3-y}{x^{2}}+\frac {\left (y^{2}-2 x \right ) y^{\prime }}{y^{2} x}=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 2] \end {align*}
type detected by program
{"exact", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
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 {y^{2} x}{-y^{2}+2 x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {-y-\frac {2 x}{y}}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (y^{2}+2 x -3 y \right )}{-y^{2}+2 x}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (y^{2}+2 x -3 y \right )^{2}}{x \left (-y^{2}+2 x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{y^{2}+2 x -3 y}\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 {2 x^{2} y^{2}-3 x \,y^{3}-6 x^{2} y +9 x \,y^{2}}{-x \,y^{2}+2 x^{2}} \\ \frac {dS}{dR} &= \frac {1}{3 R} \\ \end{align*}