Page 196 - Foundations Of Differential Calculus

9. On Differential Equations 179
through powers of dx and then completely removed through division. For
example, if the given equation is
3 2 2 2 2 3
xy d y + x dy d y + y dx d y − xy dx =0,

in which dx is taken as constant, we let dy = pdx, dp = qdx, and dq = rdx,
so that the equation becomes
2 2
xyr + x pq + y q − xy =0.
3
After the whole equation is divided by dx , this finite equation determines
the relationship between x and y.
300. Every differential equation, no matter of what order, by means of
the substitutions

dy = p dx, dp = q dx, dq = r dx,

etc., can be reduced to finite quantities. Indeed, if the differential equation
is of the first order, so that it contains only the first differential, by means
of this reduction, besides y and x, the quantity p is also introduced. If the
differential equation is of the second order, containing a second differential,
then also the quantity q is introduced; if it be of the third order, then we
also have r; and so forth. Since the differentials are eliminated from the
calculation in this way, the question about a constant differential has not
gone away. Even though we have the quantities q and r arising from second
differentials we still have to indicate whether some differential is taken to
be constant. It comes to this, whether or not in the development some
differential has been arbitrarily taken to be constant.

301. If some differential equation of second or higher order is given, and no
constant first differential is indicated, we can explore in the following way
whether or not there is a determined relationship between the variables
x and y. Since no differential is assumed to be constant, we are free to
choose whatever differential we want to be constant. By choosing different
differentials to be constant we see whether the same relationship between
x and y is given. If this does not happen, then it is a certain sign that the
equation expresses no determined relationship, and therefore can have no
place in the solution of a problem. However, the safest method, and also
the easiest, to explore this question is that given above in paragraph 277.
There, in a similar question, we gave a test for determining whether or not
differential expressions of higher order have a fixed signification.

302. Hence, given a differential equation of second or higher order, with
no differential set constant, we let dx be constant. Then, as we have shown
above in paragraph 276 for differential expressions, this equation will be
reduced to the same form, which supposes that no differential is constant.