4. (Fall 2016 Midterm) Suppose Romeo and Juliet's love affair is modeled with the following syst

So we're given this whole backstory with Romeo and Juliet. And then finally given these differen

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4. (Fall 2016 Midterm) Suppose Romeo and Juliet's love affair is modeled with the following system of differential equations: $\dot{R} = aR + bJ$ $\dot{J} = cR + dJ$ where R(t) denotes how much Romeo loves Juliet, J(t) denotes how much Juliet loves Romeo, and a, b, c, d are constants. Let $x = \begin{pmatrix} R \ J \end{pmatrix}$ be the state vector. Then the system can be represented in matrix form as $\dot{x} = Ax = \begin{pmatrix} a & b \ c & d \end{pmatrix} x$. Let $a = d = -1$ and $b = c = 1$. (a) Determine the eigenvalues and eigenvectors of A. (b) Determine $\Phi(t) = e^{At}$. (c) Let the initial condition be given as $x(0) = \begin{pmatrix} 10 \ 0 \end{pmatrix}$. This means that at $t = 0$, Romeo loves Juliet a lot, but Juliet feels neutral towards Romeo. Find the solution $x(t)$. Is the system stable? What are the values of R(t) and J(t) as $t \to \infty$?

          4. (Fall 2016 Midterm) Suppose Romeo and Juliet's love affair is modeled with the following system
of differential equations:
$\dot{R} = aR + bJ$
$\dot{J} = cR + dJ$
where R(t) denotes how much Romeo loves Juliet, J(t) denotes how much Juliet loves Romeo, and
a, b, c, d are constants. Let $x = \begin{pmatrix} R \ J \end{pmatrix}$ be the state vector. Then the system can be represented in
matrix form as $\dot{x} = Ax = \begin{pmatrix} a & b \ c & d \end{pmatrix} x$. Let $a = d = -1$ and $b = c = 1$.
(a) Determine the eigenvalues and eigenvectors of A.
(b) Determine $\Phi(t) = e^{At}$.
(c) Let the initial condition be given as $x(0) = \begin{pmatrix} 10 \ 0 \end{pmatrix}$. This means that at $t = 0$, Romeo loves
Juliet a lot, but Juliet feels neutral towards Romeo. Find the solution $x(t)$. Is the system stable?
What are the values of R(t) and J(t) as $t \to \infty$?

4. (Fall 2016 Midterm) Suppose Romeo and Juliet's love affair is modeled with the following system
of differential equations:
Ṙ = aR + bJ
J̇ = cR + dJ
where R(t) denotes how much Romeo loves Juliet, J(t) denotes how much Juliet loves Romeo, and
a, b, c, d are constants. Let x =
< p m a t r i x > be the state vector. Then the system can be represented in
matrix form as ẋ = Ax =
< p m a t r i x >
x. Let a = d = -1 and b = c = 1.
(a) Determine the eigenvalues and eigenvectors of A.
(b) Determine Φ(t) = e^At.
(c) Let the initial condition be given as x(0) =
< p m a t r i x >. This means that at t = 0, Romeo loves
Juliet a lot, but Juliet feels neutral towards Romeo. Find the solution x(t). Is the system stable?
What are the values of R(t) and J(t) as t →∞?

Added by Alex S.

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