> So the author is right, under this interpretation.

Huh?

You can say “state space,” invoke the general solution to inhomogeneous ODEs, and write it out with integrals and matrix exponentials, and the answer is still quadratic. Unless you consider t^n e^(0t) to be exponential. (Hint: what are the eigenvalues of A? One could start by calculating A^2.)

Which leads to one of the most important lessons from all of physics: you can take a problem, use a different technique to solve it, and you get the same answer! It’s magic.