Page 203 - Foundations Of Differential Calculus

186 9. On Differential Equations
2
It follows that when we multiply by R , we obtain
∂R ∂P ∂R ∂Q
P − R = Q − R ,
∂y ∂y ∂x ∂x
where the denominators ∂y and ∂x indicate that in the differentials of the
numerators, only that quantity that appears in the denominator is assumed
to be variable. However, these differentials ∂P, ∂Q, and ∂R cannot be
known until the proper value is substituted for z; since this is not known,
we proceed in the following way.

315. Since P, Q, and R are functions of x, y, and z,welet

dP = αdx + βdy + γdz,
dQ = δdx + dy + ζdz,
dR = ηdx + θdy + ιdz,

where α, β, γ, , and so forth, denote those functions that arise from differ-
entiation. Now let us consider the substitution everywhere for z the func-
tion in x and y that is equal to z, and for dz we substitute the expression
pdx + qdy and thus obtain

dP =(α + γp) dx +(β + γq) dy,
dQ =(δ + ζp) dx +( + ζq) dy,
dR =(η + ιp) dx +(θ + ιq) dy.

From these equations we obtain

∂R ∂R
= θ + ιq, = η + ιp,
∂y ∂x
∂P ∂Q
= β + γq, = δ + ζp.
∂y ∂x

316. Since the reality of the equation requires that

∂R ∂P ∂R ∂Q
P − R = Q − R ,
∂y ∂y ∂x ∂x
the result is that when we substitute the values just found, we obtain

P (θ + ιq) − R (β + γq)= Q (η + ιp) − R (δ + ζp) .

However, we have already seen that p = −P/R and q = −Q/R. But these
values can be obtained, even if the function in x and y is not substituted for