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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
6-46 Diff erential Equations
We get
11
2() () ()
pp p
J x J x dx J x dx
−+
=−
∫∫
or
11
() () 2 ()
ppp
J x dx J x dx J x
+−
=−
∫∫
6.2.14 Expansion of Sine and Cosine in Terms of Bessel
Functions
Example 6.2.8 Show that
(i)
024
cos 2 2xJ J J=− +
(ii)
135
sin 2( ).xJJJ=−+−
Solution We know that
()
1
2
1
0
1
0
1
2
01 2
2
0
1
() () () ()
() [ () ()]
11
() () ()
() ( (1) ) ()
x
t
t
nn n
nn n
n
nn
nn
n
nnn
n
n
e tJx tJx Jx tJx
Jx tJ x tJx
Jx t Jx t Jx
tt
Jx t t Jx
∞−
=−∞ −∞
=
=
==++
=+ +
⎛⎞
=+− ++ +
⎜⎟
⎝⎠
=++−
∑∑
Put
i
te
θ
=
()2 2 2
01 2
sin
01 2
()( )
(2 sin ) (2cos2 )
ii
xe e i i i i
ix
eJeeJeeJ
eJiJ J
θθ
θθ θ θ
θ
θθ
−−−
=+ − + + +
⇒=+ + +
Separating the real and imaginary parts we have
024
135
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Publisher Resources

ISBN: 9781299487406Publisher Website