4 Notes, things to find out

1. Given an ODE \(y^{\prime }\left ( x\right ) =\omega \left ( x,y\right ) \) then we want to find nontrivial Lie symmetry. The condition for this is that

   \[ \eta \left ( x,y\right ) \neq \xi \left ( x,y\right ) \omega \left ( x,y\right ) \]
   so any values for \(\eta ,\xi \) must satisfies the above.
2. Can we always find \(\xi ,\eta \) for non-trivial symmetry for first order ODE?  When I tried some in Maple, it could not find symmetries for some first order ODE’s. How does one check if nontrivial symmetry exist before trying to find one?  For example \(y^{\prime }+y^{3}+xy^{2}=0\) which is Abel ode type, Maple found no symmetry using all methods.