Page 201 - Foundations Of Differential Calculus

184 9. On Differential Equations
We must very carefully separate these kinds of equations into imaginary
and real, especially in integral calculus, since it would be ridiculous to seek
an integral for a differential equation, that is, a finite equation that it would
satisfy, when it is clear that none exists.
310. In the first place, then, it is clear that every differential equation
with three variables in which only two differentials occur is imaginary and
absurd. Let us consider an equation that contains the variable z but only
the differentials dx and dy, the differential dz being completely absent.
It is obvious that no function of x and y can be exhibited that can be
substituted for z to produce an identical equation. Indeed, the differentials
dx and dy can in no way be removed. In these cases there is absolutely
no satisfactory finite equation, unless perhaps it is possible to assign a
relationship between x and y that persists no matter what z might be. For
example, in the equation
zdy − zdx = ydy − x dx,

which is satisfied by y = x. It is easy to investigate the cases in which this
happens by looking for a relationship between x and y, first when z =0,
and then whether the same relationship persists when z has an arbitrary
value.

311. Nor is it the case that an equation is absurd only if it involves three
variables and two differentials. It can be absurd even if all three differentials
are present. In order to show this let P and Q be functions of only x and
y and consider the equation

dz = Pdx + Qdy.
If this equation is not to be absurd, then z is some function of x and y whose
differential is dz = pdx + qdy so that P = p and Q = q. However, we have
demonstrated (paragraph 232) that pdx + qdy cannot be the differential
of any function of x and y unless
∂p ∂q
= .
∂y ∂x
Here the notation means, as we previously assumed, that ∂p/∂y is the
differential of p with only y variable, divided by dy, and ∂q/∂x is the
differential of q, with only x variable, divided by dx. Hence, the equation
dz = Pdx + Qdy cannot be real unless
∂P ∂Q
= .
∂y ∂x

312. Absolutely the same reasoning applies to the equation
dZ = Pdx + Qdy,