3.2.1 Short Methods for Finding the Particular Integrals
in Special Cases
Let
0
()()
n
nr
r
r
fDyaDyQx
−
=
⎛⎞
==
⎜⎟
⎝⎠
∑
(3.37)
be a non-homogenous linear equation, where a
r
are real constants.
We will now consider short methods for fi nding the P.Is in special
cases when Q(x) is of the form (1)
aX
e
(2)
sin or cosaxax
(3)
m
x
(4)
aX
eV
and (5)
,
xV
where V is a function of x.
(1) Q(x) =e
ax
where ‘a’ is a constant
Case (i)
()0fa≠
Since
22
,,
axaxaxax
D
eaeDeae==
and generally,
raxrax
Deae=
00
()()
nn
axnraxnraxax
rr
rr
fDeaDeaaefae
−−
==
⎛⎞⎛⎞
====
⎜⎟⎜⎟
⎝⎠⎝⎠
∑∑
(3.38)
Applying
1
()fD
on both sides of Eq. (3.35)
(
)
⎛⎞
==
⎜⎟
⎝⎠
⎛⎞
⇒=⇒
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