Quiz II
1. a) The differential form
ω
= (ln
y
+
y
x

1)
dx
+ (ln
x
+
x
y

1)
dy
is exact in any
rectangle contained in the first quadrant (where
x >
0 and
y >
0). Find a function
F
(
x, y
) defined in the first quadrant such that
dF
=
ω
.
Solution
. Following the standard procedure
F
(
x, y
) =
Z
(ln
y
+
y/x

1)
dx
+
h
(
y
) =
x
ln
y
+
y
ln
x

x
+
h
(
y
)
∂F
∂y
=
x/y
+ ln
x
+
h
0
(
y
)
So we need
h
0
(
y
) =

1, which gives
h
(
y
) =

y
+
C
and
F
(
x, y
) =
x
ln
y
+
y
ln
x

x

y
+
C.
The level curves of the function
F
(
x, y
) that you found in part a) are solutions
to a first order differential equation. Write down that differential equation.
Solution
. We have
ω
= 0 when
y
=
y
(
x
) is a solution to the differential equation.
So
0 = (ln
y
+
y/x

1)
dx
+(ln
x
+
x/y

1)
dy
dx
dx
= [(ln
y
+
y/x

1)+(ln
x
+
x/y

1)
dy
dx
]
dx
So the differential equation is
0 = (ln
y
+
y/x

1) + (ln
x
+
x/y

1)
dy
dx
or in normal form
dy
dx
=

ln
y
+
y/x

1
ln
x
+
x/y

1
.
Of course, you could get that just by setting
ω
= 0, dividing by
dx
and then
solving for
dy
dx
. That “dividing by
dx
” step does not make sense, but the notation
is designed so that it leads to the right answer!
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 Spring '07
 staff
 Differential Equations, Equations, Derivative, Differential form, De Rham cohomology

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