For this problem we consider a conflict between two armies of $x$ and $y$ soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and $t$ represents time since the start of the battle, then $x$ and $y$ obey the differential equations $$\frac{dx}{dt} = -ay, \qquad \frac{dy}{dt} = -bx,$$ where $a$ and $b$ are positive constants.

Consider the case of a battle for which $a = 0.04$ and $b = 0.01$, and that the armies start with $x(0) = 59$ and $y(0) = 17$ thousand soldiers. (Use units of thousands of soldiers for both $x$ and $y$.)

(a) Using Lanchester's square law, $0.04 y^2 - 0.01 x^2 = C$, find the equation of the trajectory describing this battle; solve for the number of soldiers $x$:
$x =$
(You will need to solve for $C$ using the initial sizes of the armies.)

(b) Assuming that the army $y$ fights without surrendering until they have all been killed, how many soldiers in army $x$ does this model predict would be left when the battle ended?
$x =$