Math. 224. Test 1
Fall 2003. Oct.2
David Gurarie
Name
1.
Modeling.
Set up di
ff
erential equation models of the following systems.
Classify
them (DE/DS,
fi
rst or higher order, linear/nonlinear, autonomous, separable, etc.).
Indicate solution method(s) for each one (analytic, numeric), but
do not solve
.
(a) The logistic growth model with the growth rate
k
, carrying capacity
N
= 6, initial
population
y
0
, and harvesting rate
b
(
t
). Which
b
allow analytic solutions?
(b) Mixing problem with incoming rate
r
1
= 3
m
3
/s
, concentration
α
1
=
.
1
mg/m
3
,
outgoing rate
r
2
= 2
m
3
/s
, and initial volume
V
0
= 20
m
3
Problem
Score
1(25)
2(25)
3(25)
4(25)
Total
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(c) Financing loan model with interest rate
α
, and payment rate
p
. Write solutions,
and sketch solution curves.
Extra
: explain the minimal payment rate,
fi
nd total
payment and the overpay factor.
(d) Oscillator (massspring) system of mass
m
, friction coe
ﬃ
cient
α
, spring constant
k
, suspended vertically in the external gravity force
mg
Write it as a single equa
tion, and convert to a di
ff
erential system.
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 Spring '07
 Hahn
 Math, Logistic function, Logistic map, solution curves, outgoing rate r2

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