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Unformatted text preview: 2 MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 4 Assume ﬁrst that statement p is true. Then we have
∂M
∂ ∂f
∂2f
∂2f
∂ ∂f
∂N
=
=
=
=
=
.
∂y
∂y ∂x
∂y ∂x
∂x ∂y
∂x ∂y
∂x (We can interchange the partial derivatives when f (x, y ) has partial derivatives which are continuous functions. This is known as Clairaut’s Theorem.) This shows that statement q is true.
Now assume that statement q is true. We explain how to construct the function f (x, y ):
#1: Choose a function g (x, y ) according to the partial diﬀerential equation
x
∂g
= M (x, y )
=⇒
g (x, y ) =
M (σ, y ) dσ.
∂x
#2: Choose a function h(y ) according to the ordinary diﬀerential equation
y
dh
∂g
∂g
= N (x, y ) −
=⇒
h(y ) =
N (x, τ ) −
(x, τ ) dτ .
dy
∂y
∂y #3: Choose the function f (x, y ) = g (x, y ) + h(y ).
The function h(y ) is not a function of x because
y
y
∂
∂
∂g
∂N
∂2g
h(y ) =
N−
dτ =
−
dτ =
∂x
∂x
∂y
∂y
∂x ∂y
Then f (x, y ) is the desired function because ∂f
∂g
∂h
=
+
= M (x, y ),
∂x
∂x ∂x y
∂N
∂M
−
dτ = 0.
∂x
∂y ∂f
∂g
∂h
=
+
= N (x, y ).
∂y
∂y
∂y This shows that statement p is true.
Example 1. Consider the diﬀerential equation
dy
2 x + y2
=−
dx
2xy =⇒
2 x + y 2 dx + (2 x y ) dy = 0. We have the two functions M (x, y ) = 2 x + y 2 and N (x, y ) = 2 x y . This is an exact equation because we
have
∂M
∂N
∂M
∂N
= 2y
and
= 2y
=⇒
=
.
∂y
∂x
∂y
∂x
We use the algorithm above to construct the function f (x, y ). First, construct g (x, y ) by the equation
∂g
= M (x, y ) = 2 x + y 2
=⇒
g (x, y ) = x2 + x y 2 .
∂x
(We don’t care about a constant of integration.) Second, construct h(y ) by the equation
dh
∂g
= N (x, y ) −
= (2 x y ) − (2 x y ) = 0
dy
∂y =⇒ h(y ) = 0. Hence the desired function is
f (x, y ) = g (x, y ) + h(y ) = x2 + x y 2 .
Example 2. Now consider the diﬀerential equation
dy
3 x y + y2
=− 2
dx
x + xy =⇒ We have the two functions
M (x, y ) = 3 x y + y 2
3 x y + y 2 dx + x2 + x y dy = 0. and N (x, y ) = x2 + x y. This is not an exact equation because we have
∂M
= 3x + 2y
∂y and ∂N
= 2x + y
∂x =⇒ ∂M
∂N
=
.
∂y
∂x ...
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 Spring '09
 EdrayGoins
 Derivative

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