Link to actual problem [3978] \[ \boxed {x y \left (x +\sqrt {x^{2}-y^{2}}\right ) y^{\prime }-x y^{2}+\left (x^{2}-y^{2}\right )^{\frac {3}{2}}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _dAlembert]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= \ln \left (x \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}}}{x y +\sqrt {x^{2}-y^{2}}\, y} \\ \frac {dS}{dR} &= -\frac {1}{R} \\ \end{align*}