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Linear ODEs (review)

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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
3
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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