I am completely stuck on this:
The 2nd order system should be in this form: $\frac{dx}{dt}=Ax$ where A is the system matrix. $x(t) = \begin{pmatrix} 2-e^{-t} \\ 1+2e^{-t} \end{pmatrix}$ $x(t=0) =: x_0 = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$
I have to obtain the system matrix A and the transition matrix $\Phi(t)$ But I have no clue what I can do here. I feel like I have too less information to solve this.
I was experimenting with $\Phi(0) = \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$ and $x(t)=\Phi(t)x_0$ but I am stuck here.
Any help appreciated.
edit
$ \pmatrix{\frac{s+2}{s+1}\\ \frac{3s+1}{s+1}}-A\pmatrix{\frac{s+2}{s(s+1)}\\ \frac{3s+1}{s(s+1)}} =\pmatrix{1\\3} \implies \pmatrix{\frac{1}{s+1}\\ \frac{-2}{s+1}}=A \pmatrix{\frac{s+2}{s(s+1)}\\ \frac{3s+1}{s(s+1)}} \implies \pmatrix{1\\ -2}=A \pmatrix{1 + \frac{2}{s}\\ 3 + \frac{1}{s}} $