# MATH 333 – Fall 2014 – Quiz 3

August 31, 2017

Question
MATH 333 – Fall 2014 – Quiz 3
You may work with other class members on this quiz, but you may not receive
assistance from people not in MATH 333, Section 003 or 004. You must show
all of your work to receive full credit. Do not use decimal approximations unless
asked to do so. Any writing that appears on this quiz paper will be ignored; do
all your writing on other sheets of paper. If you are turning in more than one
sheet of paper and your pages are not all stapled together, you will earn a grade
of zero. Each quiz that is handed in must bear the names of exactly one, two,
or three members of the class; failure to abide by this expectation will result in
that quiz earning a grade of zero. Your work on this quiz must be handed in
by Friday, 3 October 2014 at the beginning of class. If it is turned in after that
time, it will earn a grade of zero. GOOD LUCK!

1) Recall the predatory-prey model

dR

dt = αR − βRF

(1)

dF

= −γF + δRF,
dt

where R is the prey population, F is the predator population, and α, β, γ,
and δ are positive constants. Suppose we modify this model to include hunting.
Speciﬁcally, assume the predators are hunted at a constant rate per unit time
and that the prey is hunted at a rate proportional to its population. Modify (1)
to include these eﬀects of hunting.
2) Consider the diﬀerential equation
d2 y dy

− 6y = 0.
dt2
dt

(2)

a) Find two solutions of (2) that are not scalar multiples of each other.
b) Letting v =
plane.

dy
dt ,

plot both of your solutions from part a) in the y-v phase

c) Convert (2) to a system of two ﬁrst-order diﬀerential equations.
3) Find the general solution of

dx

dt = 2x

dy

= −y + x3 .
dt

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