1550078481-Ordinary_Differential_Equations__Roberts_

450 Ordinary Differential Equations

where the positive constants r ands are the reinforcement rates of the x-force
and y-force and By and Dx are the combat loss rates for the x-force and
y-force, respectively.

a. For r = 3, s = 1, B = 2, and D = .75 use SOLVESYS or your

computer software to solve the following three initial value problems

on the interval [O, 5]:

(i) x(O) = 1.5, y(O) = l. 754

(iii) x(O) = 0, y(O) = 0

(ii) x(O) = 1.5, y(O) = 1.5

(NOTE: The numbers in this example have been scaled so that x, y,

r, ands can represent hundreds, thousands, or tens of thousands, etc.,

combatants. Interpret the time as being in days.)

b. For each initial value problem in part a. determine (i) which force

wins the battle, (ii) the time the battle is over, and (iii) the number

of combatants in the winning force at the end of the battle.

2. Use SOLVESYS or your computer software to solve the Lanchester combat
   model

dx

dt = (3 - .2t) - .lx - 2y

(9)

dy

dt = (1 + .3t) - .05y - .75x

for two conventional forces engaged in battle for the three initial conditions:

(i) x(O) = 1.5, y(O) = 1.75

(iii) x(O) = 0, y(O) = 0

(ii) x(O) = 1.5, y(O) = 1.5

In each case, determine (a) which force wins the battle, (b) the time the
battle is over, and ( c) the number of combatants in the winning force at the
end of the battle. In this engagement, the x-force is being reinforced at the

decreasing rate f(t) = 3 - .2t, their operational loss rate is .lx, and their

combat loss rate is 2y. The y-force is being reinforced at the increasing rate
g(t) = 1 + .3t, their operational loss rate is .05y, and their combat loss rate is
.75x.

3. Use SOLVESYS or your computer software to solve the Lanchester combat
   model

dx

dt = 2 - x - 2y

(10)

dy

dt = 1 - .2y - .2xy