Page 189 - Foundations Of Differential Calculus

172 9. On Differential Equations
290. If we need to remove several constant quantities from some finite
equation, we accomplish this through differentiating two or more times,
and in this way we finally obtain differential equations of higher orders in
which the constants have been completely removed. Let the given equation
be

2 2 2
y = ma − nx ,
2
from which we need to remove the constants ma and n. We remove the
first by differentiation, to obtain
ydy + nx dx =0.

From this we form the equation

ydy
+ n =0.
xdx
When we take dx to be constant, through differentiation, we have
2 2
xy d y + xdy − ydx dy =0.

This equation, although it contains no constant, still results from every
2
2
equation that has the form y = ma − nx, no matter what values may be
2
assigned to the letters m, n, and a .
291. Not only constant quantities can be eliminated by differentiation
from finite equations, but also some variables, namely that variable whose
differential is assumed to be constant can be eliminated by differentiation.
Indeed, from a given equation in x and y, let us find the value of x such
that x = Y , where Y is a function of y. Then dx = dY , and when dx is
2
taken to be constant, by differentiation we have 0 = d Y . However, if
2
x + ax + b = Y,
3
then by three differentiations we have 0 = d Y. By differentiation four times
the equation
2
3
x + ax + bx + c = Y
4
gives 0 = d Y . In all of these equations, although only one variable seems
to be present, while another variable can be missing from the equation,
still, since the differential dx is assumed to be constant, we must in reality
remember that there is some relationship to x and consider x as belonging
to the equation. Hence it should cause no surprise if frequently differential
equations of second or higher order occur in which only one variable seems
to be involved.