 #1
JessicaHelena
 188
 3
 Homework Statement

The population of foxes and rabbits on Nantucket Island has been studied by biologists. They measure the populations relative to a baseline, in hundreds of animals. (So ##x(2)=5## means that there are 500 more foxes than the baseline value, and ##y(2)=−5## means that there are 500 fewer rabbits than the baseline value.)
The biologists have established the following relationship between ##x(t)## (foxes' population) and ##y(t)## (rabbits' population): ##x' = 0.5x + y## ##y' = 2.25x + 0.5y##
Suppose that at ##t=0## there are ##100## more foxes than the baseline: ##x(0) = 1##; the rabbit population is at the baseline value, ##y(0) = 0##. What is the solution to this initial value problem?
 Relevant Equations

Characteristic Equation
lambda^2  (trA) lambda + det A
From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5  1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is ##{i}\choose{1.5}## ##e^{(0.5  1.5i)t}##.
Using the Euler's formula ##e^{iwt} = \cos(\omega t) + i\sin(\omega t)##, I get the real parts of ##x## and ##y## is given by
##{x}\choose{y}## = ##e^{0.5t}## ##{0}\choose{1.5}## ##\cos(1.5t)## + ##e^{0.5t}## ##{1}\choose{0}## ##\sin(1.5t)##
And given that $x(0) = 1$ and $y(0) = 0$, I arrived at:
##x(t) = \sin(1.5t) e^{0.5t} + e^{0.5t}##
##y(t) = 1.5\cos(1.5t) e^{0.5t}  1.5e^{0.5t}##
However, these equations turned out to be the wrong model.
Where might I have gone wrong? Any help would really be appreciated!
Using the Euler's formula ##e^{iwt} = \cos(\omega t) + i\sin(\omega t)##, I get the real parts of ##x## and ##y## is given by
##{x}\choose{y}## = ##e^{0.5t}## ##{0}\choose{1.5}## ##\cos(1.5t)## + ##e^{0.5t}## ##{1}\choose{0}## ##\sin(1.5t)##
And given that $x(0) = 1$ and $y(0) = 0$, I arrived at:
##x(t) = \sin(1.5t) e^{0.5t} + e^{0.5t}##
##y(t) = 1.5\cos(1.5t) e^{0.5t}  1.5e^{0.5t}##
However, these equations turned out to be the wrong model.
Where might I have gone wrong? Any help would really be appreciated!