ABSTRACT
Implicit differentiation of the function F (x , y)/C , with C , a constant, gives
@F
@x dx
@F
@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
dy
dx
@F
@x
@y
:
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)
@F
@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
@2F
@x@y
@2F
@y@x :
When expressed in terms of M and N, it follows from this that the differential
equation
M(x, y)dxN(x, y) dy0
is exact if
@M
@y
@N
@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.
