Link to actual problem [3397] \[ \boxed {3 y^{\prime }-\sqrt {x^{2}-3 y}=x} \]
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 [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{x^{2}}, S \left (R \right ) &= 2 \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 \sqrt {x^{2}-3 y}}{3}-\frac {x^{2}}{3}+2 y \\ \frac {dS}{dR} &= 0 \\ \end{align*}