82 _SEPAI_1ATION OUARIABLE_S_
[a”(x)-90’) + a(x)S"(y)] + [b”(X)P(y) + b(X)P”(y)] = 0 (9)
If S(y) and P(y) are functions such that
s"<y)=:x25<y> and P”(y)=iﬂ2P(y) (10)
If .S"'[,‘-’l=+KZS(y) then S(y) could be a linear combination of
exptity). exp(—Ky), sin
F309 5
32 SEPAngt'rION OF VARIABL_ES
Chapter g
X,(x) = sin(/l,x)
and the eigenvalues are given by
AnL=n7r for n=1,2,3,-~
It is not necessary to include n = 0 since X0(x) = sin(0x) is
identically 0 for all x.
2h: generalsolution to X" + 12X :30 is given
2-27 EXERCISE SOLUTIONS 27
_ éx—3r2[1(2m1l2)}
The terms inside the braces may be simpliﬁed to give
ﬂ: UL own”: {in 1
_[ ZWUZ _I 2m132}
dx name“) 3: °( 3 x3"2 '( 3
Setting x = L and substituting into [19),
qua = kzzR3_._.—{’L “em” {E
L
11 (2mm)
1
10(2m
38 SEPARATION o: VARIABLES
(10 = [0 " C2
Substituting (6) into the boundary condition (4) for a(L) gives
a0 + alL = IL — c2
Rear-ranging to solve for al and then using (7) to replace a0 gives
1 l
1
al =Z(tL—c2—ao)=z(tL-Cz—'o+02)=z('L_'0) (8)
Substituting