Implicit differentiation of the function F (x , y)/C , with C , a constant, gives


@x dx


@y dy0:

This is simply an ordinary differential equation written in differential form with the general solution F (x , y )/C. The equation can, of course, also be written in the more familiar form







When a differential equation

M(x, y)dxN(x, y)dy0

has the property that a function F (x , y) exists such that

M(x, y) @F

@x and N(x, y)


@y ,

it is said to be exact, and its general solution is F (x , y)/C . To test to see if an equation of this type is exact, we use the fact that when

the second order partial derivatives of F (x , y) exist and are continuous, they

must be such that




@y@x :

When expressed in terms of M and N, it follows from this that the differential


M(x, y)dxN(x, y) dy0

is exact if




@x :

the fact

@x M(x, y) and

@y N(x, y):

Integrating the first equation with respect to x , while regarding y as a

constant because M was obtained by partial differentiation with respect

to x , we find that

g @F@x dxgM(x, y) dx or

F (x, y)gM(x, y)dxg(y)A: In this result g (y ) is an arbitrary function of y and A is an arbitrary constant.

The introduction of the arbitrary function g (y) is necessary because under

partial differentiation with respect to x , f(y) will look like a constant.