Foundations of Applied Mathematics
MATH 4700 – Fall 2023
Homework 2
Due Thursday, September 28 at 2:00 PM
This homework has 100 points plus 20 bonus points available. Full credit will generally
be awarded for a solution only if it is presented both correctly and efficiently using the
techniques covered in the class and readings, and if the reasoning is properly explained. You
will not generally receive full credit, even for a correct answer, if your method of calculation is
substantially less efficient than what you should be able to do based on the class. Substantial
points on each problem will be associated with explicit statements concerning each concept
you are using, and explanations for any numbers introduced in your calculations. You should
simplify your expressions and answers as much as possible (either as a decimal or fraction),
unless otherwise specified.
1. (15 points) Use perturbation theory, assuming ϵ is a positive small parameter, to
obtain an approximation for a root of the equation
ε2x3 − x2 − 2εx+ 3ε2 = 0
with the second largest magnitude (absolute value). Your approximation should con-
sist of a nonzero main expression plus one nonzero expression for the leading order
correction. As always, you should also include a formal term describing the order of
magnitude of the error in the approximation. You won’t be penalized if you want to
calculate other roots, so long as you find them correctly, but you won’t get any bonus
points either.
2. (25 points) Use perturbation theory, assuming ϵ is a positive small parameter, to
obtain an approximation for a root of the equation
cosx =
2− x2 + ϵx
2 + ϵ2ex
which is the one with the largest magnitude satisfying |x| ≪ 1 (that is a small root that
is the least small). Your approximation should consist of a nonzero main expression
1
plus one nonzero expression for the leading order correction. As always, you should
also include a formal term describing the order of magnitude of the error in the ap-
proximation. You won’t be penalized if you want to calculate other roots, so long as
you find them correctly, but you won’t get any bonus points either.
3. (25 points plus 20 bonus points) The equation
dP
dt
= rP
(
1− P
K
)
+ I(t), (1a)
P (t = 0) = Pin (1b)
describes the dynamics of a population of size P as a function of time t under the logistic
growth model, which you hopefully studied in your elementary differential equations
class. r is a positive parameter denoting the net growth rate of the population at
low population values where competition for resources is negligible. K is a positive
parameter denoting the carrying capacity. Pin is the positive value of the population
size at the starting time.
The function I(t) denotes the rate of immigration into the population as a function of
time t. We will consider a model:
I(t) = λ(1 + α sin(γt)) (1c)
where λ, α, .and γ are some positive parameters (independent of time) which are
further restricted so that I(t) ≥ 0 always.
I don’t believe the model (1) can be solved exactly, especially by elementary methods.
So we will seek an approximate solution where we treat the effects of immigration as
weak compared to the internal birth-death dynamics of the population.
(a) (15 points) If the population starts at the carrying capacity (Pin = K), develop a
perturbation theory to compute an approximation for the population as a function
of time. Your answer should be in the form of a leading order nonzero main term,
plus an explicit nonzero correction term, plus an error estimate.
(b) (5 points) Sketch a graph of your approximate solution. Are there any apparent
ways in which it qualitatively differs from what you expect for the exact solution
to Eq. (1)? Explain your reasoning.
(c) (5 points) Over what time scale would you say your approximation from part 3a
remains valid? Provide a clear explanation for your answer.
(d) (20 bonus points) Repeat parts 3a to 3c for the case when Pin =
1
2
K.
4. (35 points) In baseball, a pitcher throws a baseball at a high speed uin ≈ 40 m/s
toward the strike zone at home plate which is a distance L ≈ 20 m away. We will
assume, for simplicity, that when the pitcher releases the ball (at time t = 0), the ball
2
is moving exactly horizontally. Due to gravity and drag force on the ball, however, the
ball will slow down and sink somewhat as it moves toward home plate. The methods
you will use in this problem can be easily extended to account for a small departure
of the release velocity from a strictly horizontal direction, as well as for the effects of
spin on the ball. The equations we will use to model these forces are:
Figure 1: Schematic of pitched baseball.
m
du
dt
= −γu
√
u2 + v2, (2)
m
dv
dt
= −γv
√
u2 + v2 −mg,
u(t = 0) = uin,
v(t = 0) = 0.
Besides the pitch speed uin and distance L to home plate previously mentioned, the
variables and parameters here are as follows:
• m ≈ 150 g is the mass of the baseball.
• u and v are, respectively, the horizontal and vertical components of the velocity
of the baseball.
• t is time since the release of the baseball by the pitcher.
• γ ≈ 0.01 g/cm is the friction constant of the baseball; it has the same meaning
as previously in the class. The reason why it multiplies a more complicated
expression here is that the drag force (which has magnitude γ(u2+ v2)) has to be
projected onto the horizontal and vertical directions.
• g = 9.8 m/s2 is the gravitational constant giving rise to a gravitational force mg
downward on the baseball.
3
We are interested in estimating the motion of the ball between the pitcher and home
plate. The equations (2) are too difficult to solve exactly, but we make the following
two observations which suggest the possibility of a perturbation theory calculation:
• The drag force and gravitational force only slightly affect the ball’s motion.
• The magnitude of the frictional drag force is roughly comparable (but not exactly
equal) to the magnitude of the gravitational force on a pitched baseball.
(a) (15 points) Conduct a perturbation theory calculation to calculate the first non-
trivial correction to the baseball’s velocity due to the effects of gravity and drag.
Include an error estimate.
(b) (10 points) When the ball reaches home plate, how far has the ball dropped
below the height at which the pitcher released the ball? Express your answers in
terms of the original (dimensional) parameters stated in the problem.
(c) (10 points) How much later do the combined effects of gravity and drag make
the ball arrive at home plate? (That is, roughly speaking, how much slower does
a baseball arrive when pitched on earth than if it were pitched on the moon!)
Express your answers in terms of the original (dimensional) parameters stated in
the problem.