Linear Differential Equations with Constant Coeffi cients 3-17
3.1.10 Exponential Shift
Theorem 3.1.19 If
12
12
()
nnn
n
fDDaDaDa
−−
=++++ where
i
a
are real constants, then
(
)
()
()()
xx
efDyfDey
αα
α
=−
where y is a function of x.
Proof
()()
(
)
()
xxx
xxxx
d
Deyeyey
dx
dy
eeyeyeDy
dx
ααα
αααα
αα
αα
−=−
=+−=
Apply
()D
α
−
again
()()
22
()()
xxx
DeyDeDyeDy
ααα
αα
−=−=
Repeating this r times, we have
()
()()
(
)
()
000
()
()()
()()
rxxr
nnn
rxxrxr
rrr
rrr
xx
DeyeD
aDeyaeDyeaDy
fDeyefDy
αα
ααα
αα
α
α
α
===
−=
⇒−==
⇒−=
∑∑∑
or
()
()()
xx
efDyfDey
αα
α
=−
This relation shows that the effect of shifting an exponential factor from
the right-hand side of the operator to its left side is to ...
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