1995 Putnam 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?

设 $x_{1},x_{2},\dots,x_{n}$ 可微

单个变量 $t$ 的(实值)函数满足

$$

\开始{对齐}

\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}

\结束{对齐}

$$

对于某些常量 $a_{ij}>0$。假设对于所有 $i$，$x_{i}(t)

\to 0$为$t \to \infty$。函数$x_{1},x_{2},\dots,x_{n}$

必然是线性相关的？