Modeling with Systems

Verify whether or not the given pair of functions, $(x(t), y(t))$ form a solution to the system

1. $x(t) = e^{-2t} + 2e^{5t}$, $y(t) = -3e^{-2t} + e^{5t}$

2. $x(t) = 2e^{-2t}$, $y(t) = -6 e^{-2t}$

3. $x(t) = 2e^{-2t}$, $y(t) = -5 e^{-2t}$

4. $x(t) = c_1e^{-2t} + 2c_2e^{5t}$, $y(t) = -3c_1 e^{-2t} + c_2 e^{5t}$

Verify whether or not the given pair of functions, $(x(t), y(t))$ form a solution to the system

1. $x(t) = 2e^{4t}\cos 3t$, $y(t) = 2e^{4t}\sin 3t$

2. $x(t) = 2e^{4t}\cos 3t - 3e^{4t}\sin 3t$, $y(t) = 2e^{4t}\sin 3t + 3e^{4t}\cos 3t$

3. $x(t) = 3e^{4t}\cos 3t - 4e^{4t}\sin 3t$, $y(t) = 3e^{4t}\sin 3t + 5e^{4t}\cos 3t$

4. $x(t) =c_1e^{4t}\cos 3t - c_2e^{4t}\sin 3t$, $y(t) = c_1e^{4t}\sin 3t + c_2e^{4t}\cos 3t$

A system is autonomous if $dx/dt$ and $dy/dt$ do not depend on $t$; that is, Which of the following systems are autonomous?

1. $\dfrac{dx}{dt} = 4x -3y; \dfrac{dy}{dt} = 3x + 4y$

2. $\dfrac{dx}{dt} = 4x -3y + \cos x; \dfrac{dy}{dt} = 3x + 4y - \sin x$

3. $\dfrac{dx}{dt} = 4x -3y +3tx; \dfrac{dy}{dt} = 3x + 4y - e^x$

4. $\dfrac{dx}{dt} = 4x; \dfrac{dy}{dt} = 4y$

Change each of the following second-order initial value problems into a system of equations.

Consider the SIR model where $S$ is the susceptible population and $I$ is the infected population.

1. Modify the SIR model to account for the situation where the susceptible population is reproducing at a rate proportional to the current population.

2. Modify the SIR model to account for the situation where the susceptible population is decreasing at a constant rate such as susceptibles leaving an infected a…