Solution of Differential Equations by Line Integrals
The total differential of a function φ=φ(x, y )is
dφ =∂φ
∂x dx +∂φ
∂y dy. (8 .17)When the right-hand side of this equation is set equal to zero, the resulting equation
is called an exact differential equation , and φ=φ(x, y ) = Constant is called a primitive
or integral of this equation. The set of curves φ(x, y ) = C =constant represents a
family of solution curves to the exact differential equation.
A differential equation of the form
M(x, y )dx +N(x, y )dy = 0 (8 .17)is an exact differential equation if there exists a function φ=φ(x, y )such that
∂φ∂x =M(x, y ) and
∂φ
∂y =N(x, y ).If such a function φexists, then the mixed second partial derivatives must be equal
and
∂^2 φ
∂x ∂y=∂M
∂y=My= ∂(^2) φ
∂y ∂x
=∂N
∂x
=Nx. (8 .19)
Hence a necessary condition that the differential equation be exact is that the partial
derivative of M with respect to ymust equal the partial derivative of Nwith respect
to xor My=Nx. If the differential equation is exact, then Green’s theorem tells us
that the line integral of M dx +N dy around a closed curve must equal zero, since
∫C©M dx +N dy =∫∫R(
∂N
∂x−∂M
∂y)dx dy = 0 because ∂N
∂x=∂M
∂y(8 .19)For an arbitrary path of integration, such as the path illustrated in figure 8-9,
the above integral can be expressed
∫C©M dx +N dy =∫xx 0M(x, y 1 (x)) dx +N(x, y 1 (x)) dy+∫x 0xM(x, y 2 (x)) dx +N(x, y 2 (x)) dy = 0