Link to actual problem [3169] \[ \boxed {\ln \left (y\right )=-x^{2}-\frac {x y^{\prime }}{y}} \]
type detected by program
{"exact", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y}{x}\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} y}{2}-\frac {3 \ln \left (y \right ) y}{2} \\ \frac {dS}{dR} &= \frac {2}{3 R} \\ \end{align*}