LINEAR DIFFERENTIAL EQUATIONS. 
33 
In this paper I propose to apply the formula (1.), by the aid of Mr. Boole’s 
solution above given, to the discovery of soluble forms of linear equations with 
variable coefficients ; I shall also show that by the use of the conversion of symbols, 
many forms of solution apparently incapable of interpretation may be made to give 
useful results ; and I shall point out a remarkable connection between the solutions 
thus obtained and the solutions of the same equations in the form of definite integrals. 
Amongst the cases of (4.) which are obviously and immediately interpretable, may 
be mentioned 
but, as will be afterwards shown, most cases are interpretable when <p(D) and X(D) 
assume the ordinary form, and consist only of integral powers. 
II. Application of these Theorems to the Solution of Equations. 
I proceed then to apply equations (5.), in conjunction with the original theorem (1.), 
to the solution in finite terms of forms of linear differential equations. Commencing 
with equations of the second degree, we have, by (1.), 
{ -h &D + } ( 'ipx.u} = -^x.'D'^u + (b-^px -j- 2 ^p'x) D^^ + ( c^-p/x + b\p'x + '<p''x)u 
{2D + &} {-p^x.u}— 24 'X.Du-\-{b'<px-{-^'^'^)u. 
Consequently equations included under the form 
x-p/x.D^u -j-((bx-l-2m) -p^x + 2x4''x)D m + ( ( c^x + bm) -p/x -\-{bx-\- 2m) -p'X + X'pJ'x) m = X, 
are readily soluble, the solution being 
u = (-p/x) “ ' (D^ - j- bD + c^) ““X } 
= (%px) ~'(D^-j-bD + c^)”'~^{x~'(D^-j-bD-j- c^) '""(xxpx.P ) } . 
When X or P is zero, the solution may be reduced to the simpler form, 
which will be found to be 
«= + (m- (m- 1)^^ 
m — 2 m{m + \) 
+ (»»- l )- 2 3 
a and (3 being the roots of t'^-\-^t-\-c'^=:0. 
m—2 m—S m[m+V){m + 2) 
+ 
•)} 
MDCCCXLVIII. 
