LINEAR DIFFERENTIAL EQUATIONS. 
481 
which, reduced to the most simple form, is equivalent to 
Make 
^i(D) *^2(D) 
<P(D)4>2(D)^^(D) d>2(D) 
a(D) ^^'(D) 
— <p{D)'if%D) + ^ (D) + 4^2(0) • 
Substituting this value in the last equation, after dividing both members by (p(D)d> 2 (D), 
or in other words, operating with (p(D)“’ 02 (D)“* on both members, we have 
{j:+W)}w={<P(D)02(D)}-V-X 
and 
m={^+^(D)}-*{<P(D)T^(D)}-V-‘X, 
where the value of 02 (D) has been substituted. By further reduction, 
^ (D) ■^^"(D)) “ ’ Z-f ?)(D)-X. 
The equation itself requires that T'(D), T^(D), ®(D), and ?.(D) should be rational 
functions of (D) ; and that the solution may be practicable, we must have 
>^(D) being a rational function. Also the term in the value of ^(D) requires 
that we should have 
P ¥>(D)rfD 
zJ — 0(D)?^°, 
0(D) being a rational function. These assumptions give, by taking the differentials 
of their logarithms, 
X(D)=?(D) (m+^j) , -I-CD) =?(D)'!'=(D) 
If we were to substitute these values in (26.), we should have a very complex 
resulting equation ; but by giving suitable particular values to the arbitrary func- 
tions, and perhaps by changing u into y'(D);s, giving to f{T)) a convenient form, we 
might obtain resulting equations sufficiently simple, and we might obtain some very 
general of their kind, remembering to introduce as many arbitrary constants as 
we can. 
Make 
%(D)=^(D), 0(D)=4^^(D); 
then 
X(D)=m(p(D)+(p'{T)), 4'^(D)=<p(D)(j?'k^(D)+4^"'(D)). 
With these values (26.) becomes, dropping the figure 2 on the top of 4", as being 
no longer needed, 
1'(D)»'=«+f(D)(p'l'(D)+^''(D))»'«-f(D)2^|p(D)(p'I'(D)+2'l''(D))|«=X, , (27.) 
