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

Lets say we have a spring with an external force.
We solve the second order ODE and arrive at:
$$x(t) = \dfrac{e^{-4 t}}{9} [3 \cos(4 \sqrt{3} t) - 11 \sqrt{3}) \sin(4 \sqrt{3} t)] + 2 \sin(8 t) \\ y(t) = 2 \sin(8t)$$
A plot of these functions is:

You can see that the $e^{-4t}$ becomes negligible when $t$ is large. The steady state solution is shown in blue and the steady-state plus transient is shown in red.
These transient terms in the solution, when they are significant, are sometimes called the transient solutions. In many physical problems, the transient solution is the least important part. However, there are cases where it is of major importance.
When the transient terms are negligible, only the $2\sin(8t)$ remains. This is called the steady-state term or steady-state solution, since it indicates the behavior of the system when conditions have become steady. You can see in the graph, that the steady-state solution is periodic and has the same period as that of the applied external force (in this case, that was, in this example, $F(t) = 24 \cos(8t)$).
Electrical circuits with resistors, capacitors and inductors (like an RLC circuit) look at transient and steady state quite a bit, so you might want to explore those.
Look at the book by Tenenbaum and Pollard for more examples regarding the damped and undamped frequency and why these things can become critically important.
