LINEAR DIFFERENTIAL EQUATIONS. 
45 
and the assertion is that a solution of the equation in the form of a definite integral 
is obtained by writing for <pz, and selecting the limits properly ; in fact, 
it is known that 
is a partial solution ; and as it is a known theorem, that if u — u^ solve 
c%=Oj 
' X ’ 
it is also solved by u—x^u^ if for n be written —n, we have for the complete solution 
u = kj' (z2 — — c^) ~”'z~^dz 
— kf'j^z^ — 1 y-h^^^dz + - 1 ) 
1 d/^u 
If for X be written ((1 — we obtain the solution of^ — m and 
f being properly taken in terms of a and n. 
In like manner, if we apply this mode of conversion to the more general form 
D2m+2QDm+ (^Q2+Q'_ 
of which the solution expressed symbolically is 
M = a?’"£”‘^‘(D2 — c2)’"“^{.r~‘(D2 — c2)“*0}, 
the definite integral ought to be 
u=kx^z~^'J'_^z^ dz-\-y 
and this is in fact the solution. 
The limits must be determined by verifying the equation and assigning them 
accordingly; the verification at the same time establishing in the particular cases 
the correctness of the results arrived at by the substitution. Thus, if we take the 
general form 
x(p(D)u-^'^(D)u=0, 
and its symbolical solution 
where xt=f^^dt, 
the conversion here indicated gives 
u = kf{ (pz)~^ i^^s^'^dz . 
