Question: Can one of the following functions be a solution of a first-order autonomous homogeneuous system?
(1) $x(t)=(3e^{t}+e^{-t},e^{2t})$
(2) $x(t)=(3e^{t}+e^{-t},e^{t})$
(3) $x(t)=(3e^{t}+e^{-t},t e^{t})$
(4) $x(t)=(3e^{t},t^2 e^{t})$
I know that every solution can be written as a linear combination of $t^j exp(\alpha t)$, therefore I would say that only (4) can be a solution, true?