Page 202 - Foundations Of Differential Calculus

9. On Differential Equations 185
if Z denotes any function of z, while P and Q are functions of x and y
without involving z. In order that Z might be a function of x and y,itis
necessary that
∂P ∂Q
= .
∂y ∂x
According to this criterion, any proposed differential equation given in this
general form can be judged to be either real or absurd. Hence, it is clear
that the equation zdz = ydx + xdy is real. Since P = y and Q = x,we
have
∂P ∂Q
=1= =1.
∂y ∂x
2 2
However, the equation az dz = y dx + x dy is absurd, since
∂P ∂Q
=2y and =2x.
∂y ∂x
But these are not equal.

313. In order to investigate this criterion more completely, let P, Q, and
R be functions of x, y, and z. Every differential equation in three variables,
provided that it is of the first order, is of the form
Pdx + Qdy + Rdz =0.

Whenever this equation is real, z will be equal to some function of x and
y. Furthermore, its differential will have the form dz = pdx + qdy. Hence,
if in the given equation this function of x and y is substituted for z and
if pdx + qdy is substituted for dz, then of necessity, the result will be an
identical equation 0 = 0. Since from the given equation we have
Pdx Qdy
dz = − − ,
R R
if in P, Q, and R, this function is substituted for z, then necessarily we
have
P Q
p = − and q = − .
R R
314. Since dz = pdx + qdy, from a previous demonstration we have
∂p ∂q
= .
∂y ∂x
Hence, when the function in x and y is substituted for z,wehave p = −P/R
and q = −Q/R, so that

∂p −R∂P + P∂R ∂q −R∂Q + Q∂R
= and = .
2
2
∂y R ∂y ∂x R ∂x