Dear Gavin,

in few words: three exponential functions e^t, e^2t, e^3t are linearly independent (over C) and form a basis in 3-dimensional linear space V of output signals z(t)=Ae^t+Be^2t+Ce^3t. Note that you obtain y(t) for A=B=C=1. Hence, the LINEAR operator T: input x(t)->output z(t) must have at least 3-dimensional linear space of input signals.

Best regards, Yuri

I would add my two cents. Three exponential terms in the solution mean that there are three eigenvalues (or three roots of the characteristic equation), indicating that the considered system is of the third order.

You may determine the order of the system easier in the S-domain. The order of the system id determined by the number of poles. So, in order to answer your interesting question, then let us transfer the system equation from the time domain to S-domain using Laplace transform.

Then Y(s)  = 1/(S-1)  + 1/( S-2) + 1/ (S-3)

Then this system has three poles at 1 , 2 and 3 in the S-plane. Since the poles lies in the right side of the S plane, then the system is unstable.

wish you success