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Solve the initial value problem in $\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{=}\mathbf{7}\mathbf{,}\mathbf{x}\left(0\right)\mathbf{=}\mathbf{0}$

Question: Consider the interaction of two species in a habitat. We are told that the change of the populations$\stackrel{\mathbf{→}}{\mathbf{x}}\left(t\right)\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{y}}\left(t\right)$can be moderated by the equation

$\left|\begin{array}{c}\frac{dx}{dt}=5x-y\\ \frac{dy}{dt}=-2x+4y\end{array}\right|$

,

where timeis a measured in years.

1. What kind of intersection do we observe (symbiosis, competition, or predator-prey)?
2. Sketch the phase portrait for the system. From the nature of the problem, we are interested only in the first quadrant.
3. What will happen in the long term? Does the outcome depend on the initial populations? If so, how?

Determine when is the zero state is stable equilibrium solution and give the answer in terms of the determinant and the trace of A