How can I reduce $s^2 + 2\zeta\omega s + \omega^2$ into something like $(s+a)^2 + (\omega + b)^2$

Question

I'm trying to solve an inverse laplace transform and I need to get this $s^2 + 2\zeta\omega s + \omega^2$ into something more workable.

Can anyone help?

Answer 1

I'm not sure if this is what you want, but the change of variables $$\left\{\begin{align}s &= x-y \\ \omega &= x+y\end{align}\right.$$

changes your expression into $$(1+\zeta)x^2+(1-\zeta)y^2$$

Also, this change of variables is $\mathcal C^{\infty}$ and hence will work well in the context of differential equations.