Begin2.DVI

Figure 8-9. Arbitrary paths connecting points (x 0 , y 0 )and (x, y ).

Consequently, one can write

∫x

x 0

M(x, y 1 (x)) dx +N(x, y 1 (x)) dy =

∫x

x 0

M(x, y 2 (x)) dx +N(x, y 2 (x)) dy. (8 .20)

Equation (8.20) shows that the line integral of M dx +N dy from (x 0 , y 0 )to (x, y )is

independent of the path joining these two points.

It is now demonstrated that the line integral

∫(x,y)

(x 0 ,y 0 )

M(x, y )dx +N(x, y )dy

is a function of x and y which is related to the solution of the exact differential

equation M dx +N dy = 0 .Observe that if M dx +N dy is an exact differential, there

exists a function φ=φ(x, y )such that φx=Mand φy=N, and the above line integral

reduces to

∫(x,y)

(x 0 ,y 0 )

∂φ

∂x

dx +∂φ

∂y

dy =

∫(x,y)

(x 0 ,y 0 )

dφ =φ

(x,y)

(x 0 ,y 0 )

=φ(x, y )−φ(x 0 , y 0 ). (8 .21)

Thus the solution of the differential equation M dx +N dy = 0 can be represented as

φ(x, y ) = C=Constant, where the function φcan be obtained from the integral

φ(x, y )−φ(x 0 , y 0 ) =

∫(x,y)

(x 0 ,y 0 )

M(x, y )dx +N(x, y )dy. (8 .22)