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XIX. On the Solution of Linear Differential Equations. By the Rev. B. Bronwin. 
Communicated by Samuel Hunter Christie, Esq., Sec. R.S. Sfc. 
Received February 6 , — Read April 18, 1850, 
If we consider the very different forms which the solutions of Differential Equations 
differing very little from each other frequently take, and the very different processes 
often required in each particular case to obtain the solution, we shall be led to con- 
clude that the discovery of any universal or general method of solving them must be 
a hopeless case. We cannot therefore regard particular methods, especially when 
applicable to a large number of cases, as useless speculations. The present paper 
contains the solution of several classes of these equations effected by means of general 
theorems in the Calculus of Operations adapted to each particular class. For expla- 
nation of the symbols employed, let it be observed that D is put for and that 
(p, X, T, and X denote any functions of x, the independent variable, and are the same 
as (p{x), X(x), &c. ; and in like manner (p(D), A(D), &c. will be used to denote the 
same functions of D, 
I. FIRST GENERAL THEOREM IN THE CALCULUS OF OPERATIONS. 
Let r=£*' ts=(pD-\-X. We easily verify 
(7S-\-h)u=r~’‘7ffr^u 
by substituting for r and nr, and then performing the operations indicated in the 
result. Change u into (nr-|-/c)M in the first member, and into its equal r~’‘z!fr^u in the 
second, and there results 
{7S-\-kyU = T~'‘7S^7^U . 
A repetition of this process will produce 
(nr-|- = r~ . 
In like manner similar equations will be found for higher powers. But the first gives 
u— (nr-j- A’)“V~*nrr^M. 
Change u into and we have 
or transposing, 
(nr-j- 
Now change u into {7s-\-h)~^u in the first member, and into its equal in the 
MDCCCLI. 3 o 
