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)