1.11 To shift \(\xi \) or not to shift

This section looks again on the option if we should shift \(\xi \) to zero and what effect this can have. As we mentioned above, given an ode \(\frac {dy}{dx}=\omega \left ( x,y\right ) \) and after finding the Lie symmetry tangent vectors \(\xi ,\eta \) and if \(\xi \neq 0\) then we can do the following

\begin{align*} \eta _{new} & =\eta _{old}-\omega \xi _{old}\\ \xi _{new} & =0 \end{align*}

So now we have new tangent vectors where \(\xi =0\) which simplifies the steps that follows. Here we will show two examples showing what happens if we do the shift and compare the solution when we do not do the shift.