[markisus]

I consider exp(At) to be an exponential expression in A. Depending on A, you may get a matrix whose entries are quadratic in t, as in this case. For another case, you may have trigonometric entries in t, as in the case of A = [0, 1; -1, 0].

Maybe you would personally say that neither of these cases is truly or essentially exponential, since we have more recognizable closed forms. But then you should also commit to saying exp(it) = cos(t) + i sin(t) is not truly exponential. I would find that a little strange, but to each his own.

Anyway, I agree that you get the same answer in the end, as you must.

[amluto]

I would say that a mass on a spring or a pendulum moves sinusoidally or harmonically (approximately, anyway), not exponentially. And I wouldn’t say that a ball falling through the air or sitting on the ground is moving exponentially, either, even if one could get at it via a matrix exponential.