Page 200 - Foundations Of Differential Calculus

9. On Differential Equations 183
307. Now let us consider differential equations involving three variables x,
y, and z that are of the first, second, or higher order. In order to investigate
the nature of these we ought to note that a finite equation composed of
three variables determines a relationship that one of them has to the other
two. Hence, there is defined the kind of function that z is of x and y. A finite
equation of this kind can be a solution insofar as it is clear what kind of a
function of x and y is to be substituted for z in order to satisfy the equation.
Likewise, a differential equation involving three variables determines what
kind of function one of the variables is of the others. Hence an equation
of this kind should be thought of as having been solved when the function
of two variables x and y is given that when substituted for z satisfies the
equation or renders it an identity. Thus a differential equation is solved if
either a function z of x and y is defined or a finite equation is given by
means of which the value of this same z is expressed.
308. Although every differential equation containing only two variables
always expresses a determined relationship between them, nevertheless this
is not always the case in differential equations in three variables. There exist
equations of this kind in which it is clear that there is no possibility that
some function of x and y can be substituted for z to satisfy the equation.
Indeed, if the given equation is

zdy = y dx,
it is easily seen that absolutely no function of x and y can be given that
when substituted for z makes zdy = ydx. The differentials dx and dy
cannot simply vanish. In a similar way it is clear that there is no function
of x and z that when substituted for y will satisfy that same equation. No
matter what function of x and z might be devised for y, its differential dy
contains dz, but this cannot be eliminated, since it is not in the equation.
For these reasons there is no finite equation in x, y, and z that satisfies the
differential equation zdy = ydx.
309. Hence we must distinguish among differential equations in three
variables those that are imaginary and those that are real. An equation of
this kind will be imaginary or absurd if there is no finite equation that could
satisfy it. Such an equation is zdy = ydx, which we have just considered.
An equation will be real if an equivalent finite equation can be exhibited
in which one variable is equal to a certain function of the other two. The
following equation is such an example:

zdy + ydz = xdz + zdx + xdy + y dx.

This fits with the finite equation yz = xz + xy, so that
xy
z = .
y − x