Full text: A Treatise on Differential Equations (Hauptband)

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
	        
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