2

We will have such a solution when ${\bf u}$ is an eigenvector of the matrix $$\pmatrix{-2 & 1\cr -1 & 0\cr}$$ for eigenvalue $r$. The eigenvalues are the roots of the characteristic polynomial $P(r) = (-2-r)(-r) + 1 = r^2 + 2 r + 1 = (r+1)^2$, so the only eigenvalue is $r=-1$. The equations for an eigenvector are $$-2 u_1 + u_2 = -u_1$$, $$-u_1 = -u_2$$. So the eigenvectors are the nonzero multiples of $$\pmatrix{1\cr 1\cr}$$, and the answer is ${\bf u}= c {\rm e}^{rt} \pmatrix{1\cr 1\cr}$ where $c$ is an arbitrary constant.