GE 207K
Exact Equations
September 8, 2011
Summary: A di
ff
erential equation which can be written in the form
M
(
x, y
) +
N
(
x, y
)
y
= 0
.
is exact
IF
M
y
(
x, y
) =
N
x
(
x, y
)
Once a di
ff
erential equation has been identified as exact, then there exists a function
ψ
such
that it satisfies the following two conditions:
ψ
x
=
M
(
x, y
)
,
ψ
y
=
N
(
x, y
)
.
The following steps can be taken to find the general solution to the di
ff
erential equation:
Steps:
1. Write the given di
ff
erential equation in the
STANDARD FORM
M
(
x, y
) +
N
(
x, y
)
y
= 0
.
2. Check for exactness:
If
M
y
=
N
x
⇒
exact.
(1)
If not, not exact.
3. Recall that
ψ
x
=
M
and
ψ
y
=
N
. Either integrate
M
with respect to
x
OR
integrate
N
with respect to
y
to find
ψ
. Choose which ever is easier to integrate. So:
ψ
=
M dx
+
h
(
y
)
(2)
OR
ψ
=
N dy
+
h
(
x
)
(3)
Remember to add an arbitrary
function
(not a constant) after integrating as shown
above.
!
4. To find the arbitary function
h
, di
ff
erentiate
ψ
you obtained and equating it with the
corresponding function
M
or
N
:
•
If you chose
ψ
=
M dx
+
h
(
y
)
then di
ff
erentiate
ψ
with respect to
y
and equate
the resulting expression to
N
. Find what
h
(
y
)
is by comparing the two sides.
•
If you chose
ψ
=
N dy
+
h
(
x
)
then di
ff
erentiate
ψ
with respect to
x
and equate
the resulting expression to
M
. Find what
h
(
x
)
is by comparing the two sides.
5. Solution is
ψ
=
C
.
(4)
1
c
hf, 2011
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GE 207K
Exact Equations
September 8, 2011
Examples solved in class:
Example 1
(1
−
y
sin
x
)
dx
+ (cos
x
)
dy
= 0
Step 1:
Find functions
M
and
N
and check for exactness.
M
= 1
−
y
sin
x
⇒
M
y
=
−
sin
x
N
= cos
x
⇒
N
x
=
−
sin
x
M
y
=
N
x
and therefore the equation is exact.
Step 2:
Find function
ψ
.
Note that we can integrate either
M
or
N
with ease.
So let’s
choose to integrate
M
with respect to
x
:
ψ
x
=
M
⇒
ψ
=
M dx
=
(1
−
y
sin
x dx
)
dx
=
x
+
y
cos
x
+
h
(
y
)
(5)
where
h
(
y
)
is a result of integrating the multivariable function
ψ
with respect to
x
.
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 Fall '08
 Fonken
 Differential Equations, Equations, Trigraph

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