Page 204 - Foundations Of Differential Calculus

9. On Differential Equations 187
z, since the differentials are no longer required in the computation. Hence
we have
PQι PQι
Pθ − − Rβ + Qγ = Qη − − Rδ + Pζ,
R R
or
0= P (ζ − θ)+ Q (η − γ)+ R (β − δ) .

Since the quantities β, δ, γ, η, ζ, θ, were found by differentiation, when we
use the notation given previously, we have

∂Q ∂R ∂R ∂P ∂P ∂Q

0= P − + Q − + R − .
∂z ∂y ∂x ∂z ∂y ∂x
Unless this condition is met, the original equation is not real, but imaginary
and absurd.
317. Although we have discovered this rule from a consideration of the
variable z, still, since all of the variables enter in equally, it is clear that
from a consideration of the other variables, the same expression will re-
sult. Hence, if a first-order differential equation involving three variables is
proposed, we can determine immediately whether it is real or imaginary.
Indeed, it can be put into the general form

Pdx + Qdy + Rdz

and then we investigate the value of the formula

∂Q ∂R ∂R ∂P ∂P ∂Q
P − + Q − + R − .
∂z ∂y ∂x ∂z ∂y ∂x
If this is equal to zero, then the equation is real; if it is not equal to zero,
then the equation is imaginary or absurd.
318. A given equation can always be reduced by division to the form

Pdx + Qdy + dz =0.

If R = 1, the previous criterion becomes simpler:
∂Q ∂P ∂P ∂Q
P − Q + − =0.
∂z ∂z ∂y ∂x
Whenever this expression is really equal to zero, the given equation is real;
if the contrary is true, then the equation is imaginary. After all of this has
been demonstrated, it is certain. However, a priori, it could be doubted
whether an equation is always real whenever this criterion so indicated.
Since at this time we cannot demonstrate this completely, nevertheless in