Page 205 - Foundations Of Differential Calculus

188 9. On Differential Equations
integral calculus this can be confirmed. At this time we simply affirm that
this is true and that no one should fear any danger from this, even if in the
meantime someone wishes to entertain some doubt about its truth.
319. From this criterion, in the first place, it is clear that if in the equation

Pdx + Qdy + Rdz =0

P is a function only of x, Q is a function only of y, and R is a function
only of z, then the equation will always be real. Indeed, since

∂P ∂P ∂Q ∂Q ∂R ∂R
=0, =0, =0, =0, =0, =0,
∂y ∂z ∂z ∂x ∂x ∂y
the whole expression vanishes spontaneously.
320. If, as before, P is a function of x and Q is a function of y, but only
R is a function of x, y, and z, then the equation will we real if

∂R ∂R ∂R ∂R P
P = Q , or = .
∂y ∂x ∂x ∂y Q
For example, if the given equation is

2 3
2dx 3dy x y dz
+ + =0,
x y z 6
since here
2 3
2 3 x y
P = , Q = , R = ,
x y z 6
we have
2 2
∂R 2xy 3 ∂R 3x y
= and = ,
∂x z 6 ∂y z 6
and so

∂R ∂R 6xy 2
P = Q = .
∂y ∂x z 6
It follows that the given equation is real.
321. If P and Q are functions of x and y, while R is a function of z alone,
since
∂P ∂Q ∂R ∂R
=0, =0, =0, =0,
∂z ∂z ∂x ∂y
the equation will be real, provided that

∂P ∂Q
= .
∂y ∂x