Let’s consider Romeo and Juliet again, and look at their love equations through the lens of the Phase Plane. For the three scenarios a), b), c) below (the same ones we analyzed in Section 3.2), do the following:
Create the system of equations in matrix form and use the eigenvalues and eigenvectors to find the solutions (we’ve done that in the previous section).
Identify the straight-line solutions and discuss their relationship when the initial conditions fall on one of these solutions.
What kind of equilibrium solution is \((0, 0)\text{?}\) Is it a sink, a source, or a saddle?
Juliet’s love changes by the amount of love she feels, and goes down by twice Romeo’s love; his love goes up by \(4\) times Juliet’s love, but down by \(5\) times his love for her.
Juliet’s love increases by twice the amount of love she feels, but goes down the same amount of love Romeo feels for her; Romeo’s love goes down by \(4\) times the love she feels for him, but goes up by \(5\) times the amount of his love for her.
Juliet’s love goes up by twice the amount of love she feels, and down by twice the amount of love Romeo feels for her; Romeo’s love goes down by five times the amount of love Juliet feels for him, and down by the same amount of love he feels for her.
2.
Now consider our salty tanks problems from the previous Section 3.2. For each set of tanks, identify the straight-line solutions, talk about what happens when our initial conditions start on one of these solutions, and identify the equilibrium point \((0, 0)\) as a sink, source, or saddle.
Bonus Question: How long will it take to completely flush the tanks so that no salt remains in either?
Tank A contains \(1000\) liters of brine, and tank B contains \(1500\) liters of brine. \(2000\) liters per hour of fresh water pour into tank A, and \(2000\) liters per hour flow out of tank B. Additionally, \(3000\) liters/hr flow from tank A to tank B, and \(1000\) liters/hr flow from tank B to tank A.
Tanks A and B each contain \(500\) liters of brine. \(1000\) liters per hour of fresh water pour into tank A, and \(1000\) liters per hour flow out of tank B. Additionally, \(1500\) liters/hr flow from tank A to tank B, and \(500\) liters/hr flow from tank B to tank A.
