I am trying to solve this problem:
Given the system $$x_1'=-x_2$$$$x_2'=2x_1+3x_2$$
Find the general solution and the set of initial conditions such that the solution tends to $0$ when $t$ tends to $+\infty$ and when $t$ tends to $-\infty$.
I could find a general solution by first trying to find a solution of the form $X(t)=\xi e^{\alpha}t$ with $\xi$ in $\mathbb R^2$ and $\alpha$ a real number. By assuming there is a solution of this form one gets to linear independent solutions $X_1(t)=c_1e^t\left(\begin{array}{r} 1 \\ -1 \\ \end{array}\right),X_2(t)=c_2e^{2t}\left(\begin{array}{r} 1 \\ -2 \\ \end{array}\right)$
Since the vector space of solutions is two dimensional, then any solution is of the form $$X(t)=c_1e^t\left(\begin{array}{r} 1 \\ -1 \\ \end{array}\right)+c_2e^{2t}\left(\begin{array}{r}1 \\ -2 \\ \end{array}\right)$$
I don't know what to do in the second part, any help would be appreciated.