Link to actual problem [9089] \[ \boxed {y^{\prime }-\frac {y^{{3}/{2}}}{y^{{3}/{2}}+x^{2}-2 y x +y^{2}}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= y, S \left (R \right ) &= -\frac {1}{x -y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {y}{2}+\frac {x}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {x -y}{\sqrt {y}}, S \left (R \right ) &= \ln \left (y\right )\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 {y^{{5}/{2}}-y^{{3}/{2}} x +2 x^{2} y -4 x \,y^{2}+2 y^{3}}{y^{{3}/{2}}+x^{2}-2 x y +y^{2}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}