Page 208 - Foundations Of Differential Calculus

9. On Differential Equations 191
Here we have

∂Q ∂R
P = z − y, =0, =1,
∂z ∂y
∂R ∂P
Q = x, =0, =1,
∂x ∂z
∂P ∂Q
R = y − z, = −1, =1,
∂y ∂x

so that the finite equation resulting from the criterion is z − x − y =0, or
z = x + y. When this value for z is substituted in the differential equation
we have
xdx + xdy − x (dx + dy)=0.

Since this equation is an identity, it follows that the differential equation
signifies nothing but z = x + y.

326. Since we have said that all first-order differential equations contain-
ing three variables are of the form

Pdx + Qdy + Rdz =0,

some question may arise concerning those equations in which the first dif-
ferentials are raised to the second or higher power. For example,

2
2
2
Pdx + Qdy + Rdz =2Sdx dy +2Tdx dz +2Vdy dz.
It should be noted about equations of this kind that they could not possibly
be real unless they have divisors of the previous form that make up simple
equations. Since from the given equation we have
Tdx + Vdy
dz =
R
2 2 2 2
dx (T − PR)+2dx dy (TV + RS)+ dy (V − QR)
± ,
R
it is clear that z cannot be a function or x and y, nor does dz have the
form pdx + qdy unless the irrational expression turns out to be rational.
This happens if

2 2 2
T − PR V − QR =(TV + RS) ,
or
2
PV +2STV + QT 2
R = .
PQ − S 2