Rit (Vísindafélag Íslendinga) - 01.06.1931, Blaðsíða 26

26
According to (31b) and (33a) its Laplacian transform is givenby:
d/>
cs V3 ,— = 0
dy
s=0
whence we infer that is a constant. From (34a) we
gather that the auxiliary equation is:
n
e'/x > ] cs t = o
s=0
Let «1, «2, £3. •. £n be the roots of this equation, and we get:
/i rsn
z=Ci / e,í,(dí?+C2 / e,,!idv + ... + C„ / e',xdí;
6lX f'2X í'nX \
(Ci e + C2 e + ... + Cn e ' / x
6lX
e2x
enX
or y Ci e + C2 e + ... + Cn e
which is the familiar solution of homogeneous linear differ-
ential equations with constant coefficients.
2. x2 + x ^ +(x2-n2)y=0
dx" dx
The Laplacian transform is:
W+l> 57+3'í ^ + <l-n’)í> = 0
a Kií/l + J?8—»?)"+ KaO/l + ++»?)"
whence v = —-------------------./- -----
Vi + v
while the auxiliary equations (34) are given by:
, dO/*0) dí> . QX
eX"(” “V “ Tv +n )
= ne>I,, [Kií/l + j;2—y)n— K^O^l + V+’y)"] = 0