Putnam 1995 A5

Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $t$ which satisfy

$$\begin{aligned}\frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \\ \frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} \\ \vdots && \vdots \\ \frac{dx_{n}}{dt} &= a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{nn}x_{n}\end{aligned}$$

for some constants $a_{ij}>0$. Suppose that for all $i$, $x_{i}(t) \to 0$ as $t \to \infty$. Are the functions $x_{1},x_{2},\dots,x_{n}$ necessarily linearly dependent?