Page 407 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Therefore for each value of n, we have a solution of (i) of the type
n n 2 2 2 /L 2
c t
V , x t A n cos xe
L
Hence the complete solution of (i) is
n 2 2 2 2
V , x t ' B A n cos xe n c t /L ...(xxvii)
n 1 L
Using the initial condition (xxvi), we have

n x
V x ,0 f x ' B A cos dx
n ...(xxviii)
n 1 L
If we integrate both sides w.r.t. x between the limits 0 to L, we have
L
1
' B f x dx ...(xxix)
L
0
n x
Also if we multiply both sides of (xxviii) by cos and then integrate w.r.t. x between 0 to
L
L, we have
L
2 n x
A f x cos dx
n ...(xxx)
L L
0
B, can also be written in a better way as
L
1
' B f x dx
L
0
L
1 2 x
. f x cos 0dx
2 L L
0
1
A 0
2
Hence complete solution of (i) to be given by
1 n x n r c t /L
2 2 2
V , x t A 0 A n cos e ...(xxxi)
2 L
n 1
L
2 n x
where A n f x cos dx ...(xxxii)
L L
0
Self Assessment

3. The heat equation is given by

2 u u
K
x 2 t

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