Link to actual problem [9109] \[ \boxed {y^{\prime }-\frac {x -y+\sqrt {y}}{x -y+\sqrt {y}+1}=0} \]
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^{\frac {3}{2}}-x \sqrt {y}-x^{2}+2 x y -y^{2}+2 y}{x -y +\sqrt {y}+1} \\ \frac {dS}{dR} &= 0 \\ \end{align*}