Fall 2003 Math 308/501–502
2 FirstOrder Differential Equations
2.4 Exact Equations
2.5 Special Integrating Factors
Fri, 12/Sep
c 2003, Art Belmonte
Summary
A
differential form
in
x
and
y
is an expression of the type
w
=
P dx
+
Q dy
, where
P
and
Q
are functions of
x
and
y
.
Simple forms
dx
and
dy
are called
differentials
. The differential
form variant
P dx
+
Q dy
=
0 is simply another way of writing
the differential equation
P
+
Q
dy
dx
=
0. They have the same
solutions, implicitly expressed as
F
(
x
,
y
)
=
C
. The level curves
(level sets in the
xy
plane) so defined are termed
integral curves
.
Let
g
=
[
x
,
y
]. Recall from Calc 3 that the (total) differential of a
continuously differentiable function
f
is the differential form
d f
=
E
∇
f
·
d
g
=
f
x
,
f
y
·
[
dx
,
dy
]
=
∂
f
∂
x
dx
+
∂
f
∂
y
dy
We say that a differential form
w
=
P dx
+
Q dy
is
exact
if it is
the differential of a continuously differentiable function
f
. (In this
case the DE
w
=
P dx
+
Q dy
=
0 is referred to as an
exact
differential equation
.) This is analogous (in the parlance of
vector calculus) to the vector field
w
=
[
P
,
Q
] being conservative
if and only if it is the gradient of a scalar potential function
f
; i.e.,
w
=
E
∇
f
. Accordingly, we may utilize familiar hand and machine
techniques from Calc 3 (in particular, Math 253) in this section.
Test for exactness
How can we test if a differential form
w
=
P dx
+
Q dy
is exact?
Phrased another way, how can we tell if the vector field
w
=
[
P
,
Q
] is conservative? From Calc 3, we just test to see if
P
y
=
Q
x
(under suitable hypotheses).
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 Spring '08
 comech
 Differential Equations, Equations, Factors, Derivative, Differential form, potential function, Art Belmonte

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