Fall 2003 Math 308/501–502
1 Introduction
1.3 Direction Fields
Fri, 05/Sep
c 2003, Art Belmonte
Summary
A firstorder differential equation in
normal form
is written
dy
/
dt
=
y
0
=
f
(
t
,
y
)
. Here
t
is the
independent variable
(think
of time) and
y
is the
unknown function
or
dependent variable
.
(Recall that other letters may be used for independent and
dependent variables.)
A
general solution
in this instance is a
oneparameter family
of
solutions to the differential equation. The parameter may be
designated as
C
or some other letter. The graphs of members of
this family are called
solution curves
.
If we are given an
initial condition
(IC)
y
(
t
0
)
=
y
0
, then the IC
together with the DE constitute an
initial value problem
(IVP).
Substituting information from the IC into the general solution of
the DE allows us to determine the parameter
C
and thus obtain a
particular solution
.
One meaning of the firstorder normal form
y
0
=
f
(
t
,
y
)
is “the
slope of the tangent line at a point is given by this expression.” Do
this for a bunch of lattice points in a rectangular region of the
ty
plane and you obtain a
direction field
. This is hard to draw by
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 Spring '08
 comech
 Math, single solution, direction ﬁeld, Polking

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