HOMOGENEOUS EQUATIONS.
Adopting the notion and the language of infinitesimals, the
earlier analysts described the above class of equations some
what more simply as homogeneous with respect to the
primitive variables and their differentials, i. e. with respect
to x, y, dx, dy, d 2 y, &c.
All equations of the above class admit of having their
order depressed by unity.
For if we assume x=e 6 , y = ez, we shall find by the usual
method for the change of variables,
dy dz
dx dd ^
(11).
<Py _ g fddz dz \
da? € U# 2 + dd)
(1«),
and so on. Here y is presented as of the first degree with
respect to e° which takes the place of x, while — is of the
d\
degree 0, and of the degree — 1, with respect to e 9 . And
the law of continuation is obvious. Hence, from the supposed
constitution of the given equation, it follows that on substitu
tion of these values the resulting equation will be homogeneous
with respect to e 9 , which will therefore divide out and leave
dz 3^z
an equation involving only z, , &c. That equation
will therefore have its order depressed by unity on assuming
%-p-- ( Art1 -)
Let us examine the general form of the result for equations
of the second order.
Representing the given equations under the form
f(x v & ^-0
^ \ X ’
dx ’ dx 2
we have, on substitution,
= 0.
a function of
i the required
ist proceed as
i found to be
ous equations
unity.
d y to be each
;gree — 1, &c.,
connects two