In differential equations, do transient state and steady state always go hand in hand? Do transient state and steady state go hand in hand in differential equations? Meaning, if there is a transient state, is there always going to be a steady state? 
Also, if there are neither of them, then is it always pure resonance? 
Could a DE not have any of those states?
Please explain and answer in the simplest terms. Thank you

I stick strictly to your question.
Of course you are speaking of a second order linear (ordinary) differential equation with constant coefficients $$x''+ \alpha x'+\beta x=f(t)
$$The theory says that its general solution is $$x=x_h+x_p$$ where $x_h$ is the general solution of the associated homogeneous equation and $x_p$ is any particular solution.
It is a fact that $$\lim_{t\rightarrow\infty}x_h=0$$ if and only if $\,\alpha>0$ and $\beta>0$.
In that case, with reference to the typical applications to physics where $t$ is time, one speaks of $x_h$ as the transient solution and of $x_p$ as the steady state solution.
If $\alpha=0$ , $\beta>0$ and in the Fouries series of $f$ (supposed periodic) there is a term with frequency equal to $\sqrt \beta$, then one has pure resonance (no transient solution, because $\alpha=0$, so speaking of a steady state solution is nonsense).
