Page 190 - Foundations Of Differential Calculus

9. On Differential Equations 173
292. It is particularly important to notice that irrational and transcen-
dental quantities can be eliminated from an equation by differentiation.
With regard to irrationals, by known methods of reduction irrationals can
be eliminated, and once this is accomplished, by differentiation we obtain
an equation free of any irrationality. However, frequently it can be more
convenient without the reduction to remove the irrationality by comparing
the differential equation with the finite formula, provided that there is only
one irrational quantity. If there are two or more irrational parts in the fi-
nite equation, then the differential equation is differentiated again as many
times as there are individual irrational parts to be eliminated, and hence
the differential equation will be of a higher order. In this way arbitrary
exponents and fractional exponents can also be eliminated. For example, if
2

y m = a − x 2 n ,
then after differentiation we have
m−1 2 2 n−1
my dy = −2n a − x x dx.
When this equation is divided by the finite equation we have
mdy 2nx dx
= − ,
2
y a − x 2
in which there remains no arbitrary exponent. It should be clear that a
differential equation that is free of all irrationality can arise from a finite
equation that involves an irrationality or even a transcendental quantity.

293. In order that we understand the way in which transcendental quan-
tities are eliminated by differentiation we begin with logarithms. Since the
differential of a logarithm is algebraic, this operation causes no difficulty.
Thus, if

y = x ln x,
then y/x =ln x, and by differentiation we have
xdy − ydx = dx ,
x 2 x
so that

xdy − ydx = x dx.
If there are two logarithms, then two differentiations are required. If

y ln x = x ln y,
then
y ln x
=ln y,
x