Description

Homework guidelines for writers:

(Adapted from the website of Professor Andy Ruina). To get full credit, please do these things on each homework.
MA 508
HW 1
On the top right corner, please put your group number, the names of your group members, with the writer at the top and clearly indicated. Also indicate any non-participating group members, e.g.:
Group 3
Jaromir Jagr (writer)
Sarah Jessica Parker
Michelle Wie
James Van der Beek (did not participate)
4. Your work should be laid out neatly enough to be read by someone who does not know how to do theproblem. For most jobs, it is not sufficient to know how to do a problem, you must convince others that you know how to do it. Your job on the homework is to practice this. Box your answers.
This homework covers 1. Linear ODEs (review); 2. Getting familiar with Matlab; 3. The phase line and trajectory sketching; and 4. Linear stability analysis.
These topic are covered in §2.0-2.4 in Strogatz.

1. Consider the following chemical reaction, where one chemical (A) turns into a different chemical (B) and vice versa. Suppose that the total amount of chemical is constant, that is A(t) + B(t) = C, where C is a positive constant. This reaction can be represented schematically in the following way:

where the two positive constants k+ and k− are called rate constants.
The following differential equation describes how A changes with time
(1) Recall that, in addition to this differential equation, we also have the conservation constraint A(t)+B(t) = C.
a) Solve for A(t), given A(0) = A0, with A0 being a positive constant such that A0 < C.
b) Use Matlab to check your answer for a few choices of A0, C, k+, and k−. (I have provided code that will assist you).

2. The position of an object moving in 1D (x(t)) on a damped, linear spring obeys the following differential equation
mx¨ = −bx˙ − kx (2)
where m, b, and k are positive constant representing the mass of the object, the damping coefficient and the stiffness of the spring, respectively.
a) Solve for x(t), given x(0) = x0, and ˙x(0) = v0.
b) Use Matlab to check your answer for a few choices of x0, v0, m, b, and k. (I have provided code that will assist you).

3. The following equation describes the velocity, v(t), of a relatively large object falling through a relatively inviscid medium (e.g., a baseball falling through the air)
mv˙ = −cv|v| + mg (3)
where m, c and g are positive constants representing the mass of the object, the drag of the medium, and the pull of gravity. a) draw a plot of ˙v vs. v. Label any equilibrium point(s) and indicate the stability of each. On the horizontal axis, indicate the flow direction.
b) without solving the equation, sketch v(t) as a function of t for several different initial conditions.
c) solve the equation for v(t), given v(0) = 0. (It will simplify your life to assume that v ≥ 0 to get rid of the absolute value sign. Once you have a solution, you can determine whether this is a reasonable assumption).
d) Use Matlab to check your solution. I have not provided code, but you should be able to modify the codefor problem 1.