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:

A

k

+

k

−

B

where the two positive constants k

+ and k

− are called rate constants.

The following differential equation describes how A changes with time

dA

dt = −k

+A + k

−B (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

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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 code

for problem 1.

4. Turn in a completed version of worksheet 1, which you worked on during class on September 1.

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