Link to actual problem [9252] \[ \boxed {y^{\prime }-\frac {y^{{3}/{2}} \left (x -y+\sqrt {y}\right )}{y^{{3}/{2}} x -y^{{5}/{2}}+y^{2}+x^{3}-3 y x^{2}+3 y^{2} x -y^{3}}=0} \]
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 [\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^{{7}/{2}}-2 y^{{5}/{2}} x +y^{{3}/{2}} x^{2}-2 x^{3} y +6 x^{2} y^{2}-6 x \,y^{3}+2 y^{4}+x \,y^{2}-y^{3}}{y^{{5}/{2}}-y^{{3}/{2}} x -x^{3}+3 x^{2} y -3 x \,y^{2}+y^{3}-y^{2}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}