Exact Differential Equations
Definition
: Let F(x,y) be a function of two real variables such that F has continuous first
partial derivatives in a domain D in the xy-plane.
The
total differential
of F, denoted by dF, is given by the expression:
dF(x,y)
=
!
F
!
x
(x,y) dx
+
!
F
!
y
(x,y) dy, for all (x,y)
"
D
Example
:
Given F(x,y) = x
3
sin(y) + y
2
x,
then its partial derivatives are:
!
F
!
x
=
3x
2
sin(y) and
!
F
!
y
=
x
3
cos(y)
+
2yx
they are continuous functions in the whole xy-plane.
Therefore, dF(x,y) = (3x
2
sin(y)) dx + (x
3
cos(y) + 2yx) dy
Definition
: The expression M(x,y) dx + N(x,y) dy is called an
exact differential form
or
a
conservative differential from
in a domain D if there exists a function F(x,y), called
the
potential function
, defined on D such that dF(x,y) = M(x,y) dx + N(x,y) dy for all
(x,y) in D.
Remark
:
If M(x,y) dx + N(x,y) dy is an exact differential form and F(x,y) is a potential function
then
!
F
!
x
(x,y)
=
M(x,y) and
!
F
!
y
(x,y)
=
N(x,y)
.
Definition
: If M(x,y) dx + N(x,y) dy is an exact differential form, then the equation
M(x,y) dx + N(x,y) dy = 0
is called an
exact differential equation
.
Example
:
The form 2xy dx + x
2
dy is an exact differential form, since it is the total differential of
the function F(x,y) = x
2
y.
Then the equation 2xy dx + x
2
dy = 0 is an exact differential equation.
We need a test to determine whether or not a given first order differential equation is