‘234 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
B. § 2. On the Solution of Linear Differential Equations by Series. 
Since, when u — ^u m z m6 , we have 
/o (D) u ( D ) ihi +ffDy J u . . . = 2 { (/ 0 (m) u m +f ( m)u m _ i +/ 2 (m) «»- 
it follows that the linear differential equation 
/o(D)w+/i(D)s^— h/ 2 (D)£ 2 ^... = 0 (6.) 
will be satisfied by the assumption u = 'Zu m z m6 , provided that 
/o(w)M m +/ 1 (rn)M m _i+/ 2 (m)z< m _ 2 ... = 0. ...... (7.) 
Let p be the lowest value of m, then since u p ~\, u p - 2 , &c. vanish, we have from (7.) 
fo ( V) ~ 0, whence the values ot p are determined. Ifjo have n real values, there will 
be n ascending developments of the form 
u=u p zP 6J rU p+ \z(p+ l ') <! -\-u p j r 2 Z ( ' p ~ >r2 y... ad infinitum , 
u p in each development being arbitrary, and the succeeding coefficients formed ac- 
cording to the law (7-)- 
This method fails when p has equal or imaginary values, but the following rule is 
of universal application. 
Rule. — Solve the equation /* 0 (D)w=0, and let the complete integral be 
m=AP+BQ+CR...., 
wherein A, B, C.. are arbitrary constants, and P, Q, R.. functions of 6. Substitute 
this value of u in the original equation (6.), regarding A, B, C.. as variable parameters ; 
the result will be of the form 
A'P-f B'Q+C'R... = 0, (XI.) 
A', B', C'.., being linear functions of A, B, C.., and their differential coefficients \ P, Q, R.., 
as before. The system of equations 
A' = 0, B' = 0, C'=0.... 
being then integrated by the fundamental theorem, the values of A, B, C will be deter- 
mined in the forms A = '2a m £ m6 , B ~^b n fi m6 , &c., a 0 b 0 .. being arbitrary constants* . 
The equation to be solved is 
/o(D)m+/ 1 (D)s^...+/,(D)s^=0, 
in which /^(D^^D), &c. are rational and integral combinations of D. This equation 
may be put under the form 
2{/„( D)s"'tt}=0, (8.) 
the summation extending from n = 0 to n=r. 
Now the solution of the equation i /’ 0 (D)M=O is of the form 
?«=AP+BQ+CR+&c., (9.) 
wherein A, B, C... are arbitrary constants, and P, Q, R... particular values of 
{,/ 0 (D)} -1 0. Substituting this expression in (8.), and regarding A, B, C... as variable 
* The reader may find it advantageous to look over some of the examples in which this rule is applied before 
reading the demonstration. 
