Example 8-6. Find the solution of the exact differential equation
(2 xy −y^2 )dx + (x^2 − 2 xy )dy = 0
Solution After verifying that My = Nx one can state that the given differential
equation is exact. Along the path C 1 illustrated in the figure 8-10 one finds
φ(x, y )−φ(x 0 , y 0 ) =
∫x
x 0
(2 xy 0 −y^20 )dx +
∫y
y 0
(x^2 − 2 xy )dy
= (x^2 y 0 −y^20 x)
x
x 0
+ (x^2 y−xy^2 )
y
y 0
=
(
x^2 y−xy^2
)
−(x^20 y 0 −x 0 y^20 ).
Here φ(x, y ) = x^2 y−xy^2 =Constant represents the solution family of the differential
equation. It is left as an exercise to verify that this same result is obtained by
performing the integration along the path C 2 illustrated in figure 8-10.
Area Inside a Simple Closed Curve.
A very interesting special case of Green’s theorem concerns the area enclosed by
a simple closed curve. Consider the simple closed curve such as the one illustrated
in figure 8-11. Green’s theorem in the plane allows one to find the area inside a
simple closed curve if one knows the values of x, y on the boundary of the curve.
Figure 8-11. Area enclosed by a simple closed curve.
