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Math 2700, Fall 2011, Practice Sheet for Exam 1
Find analytic solutions to the following initial value problems.
1.
dy
dt
=
ty, y
(1) = 3
2.
dy
dt
=
e
y
sin
t, y
(0) = 1
3.
dy
dt
=
e
t
y
ln
y, y
(0) =
e
4.
dy
dt
=
(sin
y
)(cos
t
)
e
y
ln(
y
+3)
, y
(5) = 0
Modeling:
5.
Suppose that we have a population of rabbits on an island. Left alone, the rate at which their
population increases is proportional to the total size of the population. Suppose also, rabbits are
removed from the island at a rate of 100 rabbits per year.
Assuming that the initial size of the population is 500 rabbits, and that after 1 year it is 700 rabbits,
ﬁnd an equation for the population
P
as a function of time.
6.
Suppose that we have a tank ﬁlled with pure water with a initial volume of 10,000 liters,
and that a solution with a concentration of 5g of salt per liter is poured in at a constant rate of 10
liters/minute. The solution is then mixed into the water and the resulting mix is removed from the
tank at a constant rate of 20 liters/minute (until the tank is empty).
Find an analytic description of the concentration of salt in the tank as a function of time (applicable
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This note was uploaded on 12/10/2011 for the course MATH 2700 taught by Professor Staff during the Fall '08 term at University of Georgia Athens.
 Fall '08
 Staff
 Math, Differential Equations, Equations

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