ODE, Homework 1
Due June 2, 2008
1.
Draw a reasonably detailed direction field for
y
=
y
+
t
. Sketch 3 solutions with different
behaviors. Are there any equilibrium solutions? Are there any asymptotic solutions? If so,
what are they?
2.
Sketch a direction field for
y
= sin(
y
).
Sketch at least 3 equilibrium solutions and at
least 4 others with different behavior. What are the equilibrium solutions? What are their
stabilities? At what values of
y
do the solutions have inflection points?
3.
Write down a differential equation of the form
y
=
ay
+
b
for
a, b
∈
R
such that they have
the required behavior as
t
→ ∞
: a) all solutions approach
y
= 26, b) all nonequilibrium
solutions diverge from
u
= 117
/
29.
4.
Write down a differential equation such that as
t
→ ∞
, all solutions approach the line
y
= 2
t
.
5.
(2.5.28 from text) Chemical reactions. A second order chemical reaction involves the in
teraction (collision) of one molecule of a substance
P
with one molecule of a zubstance
Q
to
produce one molecule of a new substance
X
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 Spring '11
 Peters
 Chemistry, Differential Equations, Equations, 2 gal, 100 gallons, 50 oz

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