solutions fort-order homog. system 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?

I assume by "homogeneous system" you mean "homogeneous linear system". 
A $2 \times 2$ matrix can either have two eigenvalues or one eigenvalue of multiplicity $2$.  In the first case the solutions are linear combinations of vectors times two exponentials $e^{\lambda_1 t}$ and $e^{\lambda_2 t}$.  That's what you have in one of your four examples.
In the second case you have one exponential and $t$ times that exponential. 
But you can't have three different $\lambda$'s, or $e^{\lambda_1 t}$ and $t e^{\lambda_2 t}$ with $\lambda_2 \ne \lambda_1$, or $t^2 e^{\lambda t}$.  
