LINEAR DIFFERENTIAL EQUATIONS. 
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simple. A concise statement of its scope when restricted to 
these is as follows : 
Having given an equation 
% + {X) .«■-.(*>§+ }».(*)»- 0, (A) 
in which p 1 (x), p 2 ( x ), . . . p n ( x) are rational fractional func¬ 
tions of x, Cauchy’s theorem asserts the existence of an inte¬ 
gral, y (x), for equation (A) at every point x 0 , when x 0 is not a 
point at which one or more of the p’s become infinite, said 
integral being expressible by a power series having the form 
yOO^o + O — + 0 — ^o) 2 ^+. (B) 
the f) s being constants. The series (B) converges for all 
values of x included in a circle described about x 0 as a center, 
provided, always, that this circle contains no point at which 
one or more of the p’s become infinite. These excluded points 
are the singular points of the function defined by equation 
(A), and are the only singular points of the function in the 
finite plane. They are fixed—that is, they are independent 
of the constants of integration. The first n coefficients of 
series (B) remain arbitrary and can be assigned at pleasure. 
We therefore have 
dy ~1 
m 
zsz l = 7 h 
that is, the first n — 1 derivatives of the said function remain 
arbitrary. 
The unique existence of a general integral of an equation 
of the form of (A) is established by the above theorem. Its 
direct calculation, however, by the method of undetermined 
coefficients is, in general, impossible. 
The general integral can be reduced to a more manageable 
form by writing it as the sum of n particular integrals each 
