Page 207 - Foundations Of Differential Calculus

190 9. On Differential Equations
Then
P = z − x, Q = y − z, and R =0,
but
∂P ∂Q
= 1 and = −1.
∂z ∂z
The deciding equation becomes

∂Q ∂P
P = Q ,
∂z ∂z
or

z − x = z − y,

so that

y = x.
Since in this case it turns out that y = x also satisfies the differential
equation, we have to say that the given differential equation has no other
significance than y = x.
324. Hence, when a differential equation containing three variables is
given,

Pdx + Qdy + Rdz =0,

there are the three following cases that must be considered concerning the
equation which results:

∂Q ∂R ∂R ∂P ∂P ∂Q
P − + Q − + R − =0.
∂z ∂y ∂x ∂z ∂y ∂x
The first case occurs if this expression is really equal to zero, and then the
given equation is real. However, if this finite equation is not an identity,
then it must be decided whether it satisfies the given equation. If this
happens, we have a finite equation, and this is the second case. The third
case occurs if the finite equation does not agree with the given differential
equation, and then the given equation is imaginary. In this case no finite
equation can be found that satisfies the given equation.

325. The first and third cases are self-evident. The second, however, al-
though quite rare, deserves special consideration. Since the example already
considered contains only two differentials, we will give another equation,
which has all three differentials:

(z − y) dx + xdy +(y − z) dz =0.