The Helmholtz wave equation is given as :
$$\nabla^2 \vec E =\mu\epsilon \frac{\partial^2 \vec E}{\partial t^2}$$
Considering $\vec E=E_x(z) e^{j \omega t}$ the Helmholtz wave equation now takes the form
$$\frac{\partial^2 E_x}{\partial z^2} + \omega^2 \mu \epsilon E_x=0$$
I know the solution to this equation can take many forms.
But which solution is generally used?
I mean some books use $$E_x=(E_x^+ e^{-jkz}+E_x^- e^{jkz})e^{j \omega t} = E_x^+ e^{j(wt-kz)}+E_x^- e^{j(wt+kz)}$$
while some other books use $$E_x=E_{0x}\cos(wt-kz)$$ where $E_{0x}$ is some value obtained from initial conditions.
Am I at liberty to use any of these solutions considering what is applicable in the problem in front of me?
I see that for formulation equations of polarization the cosine solution is used and not the exponential solution.