Page 184 - Foundations Of Differential Calculus

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On Differential Equations

281. In this chapter we principally set forth an explanation of the differen-
tiation of those functions of x that are not defined explicitly, but implicitly
by means of the relationship of x to the function y. Once this is accom-
plished, we consider the nature of differential equations in general, and
we show how they arise from finite equations. Since the main concern in
integral calculus is the solution of differential equations, that is finding fi-
nite equations that correspond to the differentials, it is necessary here to
examine very carefully the nature and properties of differential equations
that follow from their origin. In this way we will be preparing the way for
integral calculus.

282. In order that we complete this task, let y be a function of x that is
defined by this quadratic expression:
2
y + Py + Q =0.
2
Since this expression y +Py+Q is equal to zero, whatever x might signify,
the equation will still be equal to zero if we substitute x + dx for x. In this
case y becomes y + dy. When this substitution is made and the original
2
y + Py + Q is subtracted from the new quantity, there remains the dif-
ferential, which is also equal to zero. From this it should be clear that if
any expression is equal to zero, then its differential will also be equal to
zero. Furthermore, if two expressions are equal to each other, then their
2
differentials will be equal. Since y + Py + Q = 0, we also have
2ydy + Pdy + ydP + dQ =0.