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Differential Equations

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Difficult

Solutions of System Converge to Zero

DIFFEQ-EXD0CE

Consider the following system of first-order linear differential equations:

$$x_1'=-x_1+2x_2,\ x_2'=ax_1-x_2$$

...where $a$ is a positive constant and the unknowns are $x_1=x_1(t)$ and $x_2=x_2(t)$.

Find all values of $a$ such that all solutions of system converge to zero as $t\rightarrow\infty$.

A

$0 < a <\cfrac{1}{2}$

B

$0 < a\le\cfrac{1}{2}$

C

$a>\cfrac{1}{2}$

D

$a\ge\cfrac{1}{2}$