Section 3.2 Planar Systems
Exercises Exercises
Supplemental HW Problems.
1.
Let’s revisit our Star-Crossed lovers from
Section 2.2 and their systems of differential equations. For the questions below, set up the equations, find the matrix and its eigenvalues/eigenvectors, and use that information to write the general solution of Juliet’s and Romeo’s love for each other. Can we tell from the given information what will happen to their love for each other over time?
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 the three situations above but with initial conditions. Use these three initial conditions for each of the situations above: \((0, 0),\) \((1, 1),\) and \((-1, 1).\) What are the prospects for their relationship long-term with these conditions?
3.
Consider
example 3.2.1 1 from our textbook (a system of two tanks filled with brine). For the following tank scenarios a) and b), perform the eigenvalue/eigenvector analysis to find the solution, and then determine how long it will take each tank’s salt concentration to get under
\(1\) gram/liter (
\(1\) gram is
\(0.001\) kg). Do this for four different sets of initial conditions:
Tank A contains \(100\) kg salt, and Tank B contains \(0\) kg salt
Tank A contains \(0\) kg of salt, and Tank B contains \(0\) kg salt
Tank A contains \(0\) kg salt, and Tank B contains \(100\) kg salt
Both tanks start with \(100\) kg salt in each.
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.
4.
The forest behind my house has two different species – foxes and mice, in a predator-prey relationship. The more mice there are, the faster they breed, and more foxes means a slower growth rate for the mouse population; the rate of change of the mice is \(4\) times the current mouse population, but negative-five times the current fox population. Similarly, more mice means that more foxes can thrive, but more foxes means a slower growth rate as they compete for resources. The rate of fox growth is three times the number of mice, but negative-four times the number of foxes.
Present the analysis of this fox-mouse interaction with a system of differential equations, and then solve that system for the equations that give the mouse population and the fox population over time. If \(m(t)\) gives the mouse population in thousands in week \(t\text{,}\) and \(f(t)\) gives the fox population in tens in week \(t\text{,}\) discuss the long-term behavior of the fox and mice populations when the forest starts with \(2000\) mice and \(20\) foxes, \(2000\) mice and \(19\) foxes, and 2000 mice and \(21\) foxes.
faculty.sfasu.edu/judsontw/ode/html-20210811/linear02.html
