- 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.
- 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.