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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
6-24 Diff erential Equations
Equating the coeffi cients of
1
1
n
t
+
on both sides
21
0
()
(1cos)
n
n
d
Px
xx
π
θ
π
θ
+
=
±−
or
(1)
21
0
1
() ()
(1cos)
nnn
n
d
Px P Px
xx
π
θ
π
θ
−+
+
==
±−
6.1.15 Expansion of f(x) in a Series of Legendre
Polynomials
Example 6.1.13 Let f (x) be expressible in a series of Legendre poly-
nomials. Thus
00 11
0
() () () () ()
nn nn
n
fx CPx CPx CPx CPx
=
==++++

(6.64)
where C
n
are constants to be determined.
Then multiplying both sides by ()
n
Px and integrating w.r.t. x from
–1 to 1, we get
11
00 11
11
1
2
1
1
1
() () [ () () () ] ()
2
()
21
21
() ()
2
nnnn
nn n
nn
f x P x dx C P x C P x C P x P x dx
CPxdxC
n
n
CfxPxdx
−−
=++++
==⋅
+
+
⇒=
∫∫

1
1
1
2
1
() () 0 if
2
() if
21
mn
n
PxPxdx m
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Publisher Resources

ISBN: 9781299487406Publisher Website