Chapter 5.2, Problem 38E

Arms Race. A simplified mathematical model for an arms race between two countries whose expenditures for defense are expressed by the variables $x\left(t\right)$ and $y\left(t\right)$ is given by the linear system

$\frac{dx}{dt}=2y-x+a$; $x\left(0\right)=1$,

$\frac{dy}{dt}=4x-3y+b$; $y\left(0\right)=4$,

where $a$ and $b$ are constants that measure the trust (or distrust) each country has for the other. Determine whether there is going to be disarmament ( $x$ and $y$ approach $0$ as $t$ increases), a stabilized arms race ( $x$ and $y$ approach a constant as $t\to +\infty$), or a runaway arms race ( $x$ and $y$ approach $+\infty$ as $t\to +\infty$).