LINEAR DIFFERENTIAL EQUATIONS. 
463 
which is one order lower than the proposed. All equations therefore of the second 
order included in (1.) may be considered as integrated by this process. 
Another form, distinct from the above, is 
nk')u-\-pf^{'UT)'mr'‘u=l^ (3.) 
To reduce this we must assume 
Then by {a.) we have 
r^u=T^~^{vT-\-k)~^ (t<r+(w— l)^)“V^y. 
Substituting these values and operating with 1)^) on both 
members of the result, we have 
(4.) 
The observations made with reference to the former example might be repeated 
here, I shall add, that as k may be both positive and negative, these two examples 
include every variety of case of this form of equation. 
Before I proceed to notice particular examples, it may be as well to give a few 
more general ones, and thus to point out the whole series of them which are suscep- 
tible of reduction by this method. 
Let f{7!T)vj(i!y-\-k)u-\-pf^{vi){'t;y-{-nk)r^^u'='K. (5.) 
Make u—{js-\-2k){7n-\-^k) {y3-\-{n-\-\)k)v, 
the common difference of the factors being k, as before, which here also may be both 
positive and negative. This will give 
{7;y-\-in — \)k)r^’‘v, 
and proceeding exactly as before, we shall arrive at the reduced equation. 
Again, suppose 
The assumption 
leads to 
X (6.) 
(/.) 
M=(K5'-}-2A:)“'('z«7-f-3A’)~‘.... (vT -\-{n-\-\)k)~'^v 
/(^^r)(sJ-{-^)^;-l-;?/j(^3■)r"*^;=p|^” ^^^jx (8.) 
The two last examples, like the former, are reduced an order lower ; and when 
they are of the second order, they may be considered as integrated. In order to 
enable us to effect their reduction, it is necessary that we should have two operating 
factors, as ?;y(TO--{-A), in one of the terms, these factors having the difference k. I shall 
only give two more examples, which will suffice to indicate the series of them before 
mentioned. 
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