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Differential Equations by Pearson
book

Differential Equations by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
472 pages
13h 25m
English
Pearson India
Content preview from Differential Equations by Pearson
Linear Diff erential Equations with Constant Coeffi cients 3-17
3.1.10 Exponential Shift
Theorem 3.1.19 If
12
12
()
nn n
n
fD D aD aD a
−−
=+ + ++ where
i
a
are real constants, then
(
)
()
() ( )
xx
efDyfD ey
αα
α
=−
where y is a function of x.
Proof
() ()
(
)
()
xxx
xxxx
d
Dey eyey
dx
dy
eeyeyeDy
dx
ααα
αααα
αα
αα
−=−
=+−=
Apply
()D
α
again
() ( )
22
() ()
xxx
DeyDeDyeDy
ααα
αα
−=− =
Repeating this r times, we have
()
() ( )
(
)
()
000
()
() ()
() ()
rx xr
nnn
rx xr x r
rrr
rrr
xx
DeyeD
aD e y aeDy e aD y
fD e y e fDy
αα
ααα
αα
α
α
α
===
−=
⇒−= =
⇒−=
∑∑∑
or
()
() ( )
xx
efDy fD ey
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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Publisher Resources

ISBN: 9781299487406Publisher Website