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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
6-58 Diff erential Equations
The relationship between
()
n
Tx
and
()
n
Tx
is given by the recur-
rence relation
2
1
(1 ) ( ) ( ) ( )
nnn
xTx nxTx nT x
−=−+
(6.129)
Putting cosx
θ
= we get
2
2
LHS (1 cos ) cos
1
sin ( sin ) sin sin
sin
dd
n
ddx
nn n n
θ
θθ
θ
θθ θθ
θ
=−
⎛⎞
=− −=
⎜⎟
⎝⎠
RHS cos( 1) cos cos
[
cos cos
sin sin cos cos ]
sin sin
nn n n
nn n n
nn
θθθ
θθ θθ θ θ
θθ
=−−
=+−
=
Since
LHS RHS=
Eq. (6.115) is proved.
Example 6.3.1 Write the follo
wing polynomials in terms of Chebyshev
polynomials of fi rst kind:
(a)
2
257xx−+
(b)
32
81134xxx+−+
(c)
432
16 8 2 4 3xxxx−−+−
Solution We have
23
01 02 13
11
1 , , ( ), (3 ),
24
TxTx T T x TT===+ = +
4
024
1
(3 4 )
8
xTTT=++
2
02 1 02 1 0
32
13 02 1 0
3210
43
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Publisher Resources

ISBN: 9781299487406Publisher Website