LINEAR DIFFERENTIAL EQUATIONS. 
47 
To verify this and find the limits, we have 
=fa"(z— i«)*(z-/3)“...s»' — + 12 “"‘+ — (3)'- 
s"»‘ (h~'‘ 
bji 
i'"" dz 
= — iD” + • • + ^iDm + Z> o), 
if — a)'^(z— (3)®. can be made to vanish between the limits; and this condi- 
tion is satisfied if — oc be taken as the lower limit, and a, jS, &c. be taken successively 
as the upper limit, whence the complete solution. 
If the expression . has m roots equal to a, the form of the solution will be 
modified. If in such case the rational fraction 
bn^ + ... _b„ A,„ A ^_1 4 _ 4 _ 
a,iZ”' + ... (^2 — z — a.' z — /3 ' 
the solution becomes 
u 
1 , Am -1 1 
m—\ _ ,w _2 — 
{^—Kj 771 — Z (z — a) 
"(z — ay‘(z — (3y 
in 
San'^S^'^dz 
between the limits — oc and a, (3, See. This solution is incomplete ; but it may be 
completed by using instead of a„ 2 ”+. . .its first, second, (m— l)th differential co- 
efficients. 
There can be no doubt that this remarkable connection between the symbolical 
solution and the solution by definite integrals is not merely accidental, but is founded 
upon a similarity in the processes by which they would be respectively arrived at in 
a general system of solution. 
The following considerations are offered as in some measure explanatory of the 
connection above adverted to. The equation to be solved is of the form 
.r ( D ) M -f ( D ) M — 0 . 
Now if u—kP ‘mz.i'^'dz, we have 
n. 
9(D) M = 'uyz.(pz.z‘''dz = ^e‘“'rnz.(pz 
between the limits. And 
y,(Y})u=:k r Z!TZ.'<Pz.S^‘'^dz. 
^ a 
If the limits be— oc and a root of 9;s=0, the equation is verified if njz.'^z= ■^(z^z.<pz), 
which requires 
log 
