LINEAR DIFFERENTIAL EQUATIONS. 
37 
that the expressions (5.) give, suppressing h, 
I I I 7 .Gm4^'x , ^rnyx V' A p 
a=(+j)-‘(D3+cD+/)”-{j-'(D’+cD+/)-“(^4^.P)} ; 
or 
D3K+3QD%+(c+3(Q2+Q')-3’?fi5^)DK+(/+cQ+QHQ''+3QQ' 
M=ir’"r‘‘'(D3+cD+/)“-'{a:-'(D3+cD+/)-”(.r-”+'£«‘P)). 
And the expressions (6.) give 
D3„+(j+^+3tf)D3«+(.+^+2(6+^)^^^+2f)D. 
, f m f bm\^'x /, m\4/"x \p"'x\ „ 
M = (-v^.r ) “ ‘ (D^ + 6D + c) “ ‘D”- P) 
It is obvious, however, that the generality of the soluble forms becomes less, as the 
order of the equation rises. 
The solutions derived from (5.) and (6.) as particular forms of (4.), have been given 
in the first instance on account of their peculiar simplicity; but more general forms 
are derived by the use of (4.). 
The expressions (4.) represent the solution of linear equations of any order, in the 
factors of which no power of a? higher than the first appears. 
The general form is 
'M + ... + (fli^-l- 6 j)De/ + (aoJ’-f &o)m=X, . . ( 10 .) 
in which 
\}/f= + .. + ^^# + 60 
lp^ = aX+®n-y”~' + --- + «l^ + «0 
r^f{t+t=-.+T=-^+t=-^+--yt=ft+ log ■). 
where a, (3, y, See. are the roots of (pb=0 ; and A, B, C, &c. are found from the rational 
fraction Consequently the solution of this equation is 
u=(a„D»+a„_iD”-+..+ aiD+ao)-V»''(D-a)^(D-|3)^...(^a^-^{s-r„'’(D-a)-^(D-|3)-^. 
The factor denotes that x is to be changed into x —— ; and the factor de- 
notes that x is to be changed into ; but these factors may be dispensed with by 
making 6„=0, whieh does not diminish the generality of the form. 
