Since the line integral is independent of the path of integration, it is possible to
select any convenient path of integration from (x 0 , y 0 )to (x, y ).
Figure 8-10. Path of integration for solution to exact differential equation.
Illustrated in figure 8-10 are two paths of integration consisting of straight-line seg-
ments. The point (x 0 , y 0 )may be chosen as any convenient point which guarantees
that the functions Mand Nremain bounded and continuous along the line segments
joining the point (x 0 , y 0 )to (x, y ).If the path C 1 is selected, note that on the segment
AB one finds y=y 0 is constant so dy = 0 and on the line segment BE one finds xis
held constant so dx = 0. The line integral is then broken up into two parts and can
be expressed in the form
∫(x,y)
(x 0 ,y 0 )
dφ =φ(x, y )−φ(x 0 , y 0 ) =
∫x
x 0
M(x, y 0 )dx +
∫y
y 0
N(x, y )dy, (8 .23)
where xis held constant in the second integral of equation (8.23). If the path C 2 is
chosen as the path of integration note that on AD xis held constant so dx = 0 and
on the segment DE yis held constant so that dy = 0. One should break up the line
integral into two parts and express it in the form
∫(x,y)
(x 0 ,y 0 )
dφ =φ(x, y )−φ(x 0 , y 0 ) =
∫y
y 0
N(x 0 , y )dy +
∫x
x 0
M(x, y )dx, (8 .24)
where yis held constant in the second integral of equation (8.24).
