Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 71
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a(Ðx—a/1 (Ðx—o.2)^2(Ðx— a3)@3 ....-*y = z(x) (ld)
or according to (lOb)
acchxÐgi(e(ÐP'(e(a*-«*'>xÐps(....Ðtít(e - asx y))))=z(x)
(11)
The factors (Ðx—ar)+ of (1 d) are commutative and, therefore,
ea<xÐxpi(e~“j1 in the sequence on the left-hand side of (11)
are also commutative. Solving (11) with repect to y we can,
therefore, write
y — --ea'xÐ~Pi(e(a%-a')xÐ~&2(.... Ð~P* (e~asxz))) (12)
Example:
%+<b-2a > S + a<“-2*) fz+a'b,,=1
or
Hence
(Ðx + b)(Ðx —af(y)=x
y=e~bx Ð~l (e{a+b)x Ð ”2 (e~axx))
“-“S’ ein+h)x dx j dx j e~axxdx
=C ixeax +Cieax +C3e~bx +
C2 C3
X
a2b
2 b—a
a3b2
where
— ■aCi. 1 1 e~acs
1 a2(a+6)
C2 =
(ac3+l) (a+b)cz+1
a2(a+bY
ac3_______
^ Uto _____
0(72+2
a3(a+6)
e
, 6ci
a—-2b —abcj (ac3+\)[(a+b)(ci—c2)—1]
,n(ci - c.)
+ i
GC2 + 2
L a3 b2 a2(a+6)2 a3(a+í>)
It is easy to see from this integration that y becomes o for
x=C\. If ci=cz, then we havej=0 and ^- = 0 f°r x = Ci.
dv d2y
Moreover, if c3=Cí=ci then is y=0, ^x =0 ancl
for jc=ci .
,a(ci —C2)