Link to actual problem [3979] \[ \boxed {\left (x \sqrt {1+x^{2}+y^{2}}-y \left (x^{2}+y^{2}\right )\right ) y^{\prime }-x \left (x^{2}+y^{2}\right )-y \sqrt {1+x^{2}+y^{2}}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= x^{2}+y^{2}, S \left (R \right ) &= -\arctan \left (\frac {x}{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 {x^{2} \sqrt {x^{2}+y^{2}+1}+y^{2} \sqrt {x^{2}+y^{2}+1}}{-x^{2} y -y^{3}+x \sqrt {x^{2}+y^{2}+1}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}