LINEAR DIFFERENTIAL EQUATIONS. 
41 
o C 
A^ain, if a——-, then equation becomes 
^~^(^+-)(r^^2M=Ro ; 
whose solution, therefore, is 
«^^D-(« + i) { (^,2q_c2)-(i+t)D-i { (^2.}. ^.2)4+iDi(^2q_c2)-«RJ } } . 
Thus the solution of 
d'^u 3 
^\ + f + k't^ 
IS 
u 
n+t^+hlt\ 
= P. 
The principle illustrated in this section may be further exemplified and usefully 
applied, by attempting the solution of the general equation of the second order, 
(px.T)‘^u + \px.Da -}- )^x.u=P. 
By the interchange of symbols, we have 
(p(D)(x^u) - ■<Jy(D) (xu) -j-x(D)u=P ; 
or x^(p(D)u-}-2x^'(D)u-l-^"(D)u' 
—x\P(D)u—'^'(D)u 
+%(D)m 
This equation is soluble if %(D)m=:{-Jv'(D) — ^"(D)}m ; for it then assumes the form 
x(p(D)u-j-(2ip'(D) — ’<fy(D))u=iV~'P ; 
the solution of which, by (4.), is 
u=(p(D)s~^f(D)‘^^'lx~'^(fi(D))~^£'^^w‘^^{x~^P} } ; 
and, by restoring the symbols, we get for the solution of 
.D^m + •<px.Du -h — <P''^) u=P, 
(P) } , 
the correctness of which may be ascertained by verification. 
If '4'X=(n-i-l)fi'x, we have for the solution of 
^x.D^u-p(n-j-l)(p'x.Du-i- n(p"x.u = P, 
u= (cpx) ~^<pxY~^Pdxdx. 
If be made equal to Q(px, and P to P(px, we get for the solution of 
D2«+QD„+(q’+Q^^-^)„=R, 
MDCCCXLVIII. 
<px fx , 
u=(px.r^'D~^{((px)~h^'D~'(R<px)}. 
G 
