Link to actual problem [8991] \[ \boxed {y^{\prime }-x^{2} \sqrt {x^{2}-2 x +1+8 y}=-\frac {x}{4}+\frac {1}{4}} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
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 &= 2 x, \underline {\hspace {1.25 ex}}\eta &= x^{2}-\frac {5}{2} x +12 y +\frac {3}{2}\right ] \\ \left [R &= \frac {x^{2}-2 x +1+8 y}{8 x^{6}}, S \left (R \right ) &= \frac {\ln \left (x \right )}{2}\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 &=-2 x^{3} \sqrt {x^{2}-2 x +8 y +1}+\frac {3 x^{2}}{2}-3 x +12 y +\frac {3}{2} \\ \frac {dS}{dR} &= 0 \\ \end{align*}