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): \begin{align} x' &= 0.5x + y \\ y' &= -2.25x + 0.5y \end{align}
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?
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 $\binom{i}{1.5}$. Hence the complex solution is $\binom{i}{1.5}e^{(0.5 - 1.5i)t}$.
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
$$\binom{x}{y} = e^{0.5t} \binom{0}{1.5} \cos(1.5t) + e^{0.5t}\binom{1}{0} \sin(1.5t).$$
And given that $x(0) = 1$ and $y(0) = 0$, I arrived at: \begin{align} 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} \end{align}
However, these equations turned out to be the wrong model.