Page 397 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
u w

v w
i.e.,
x v w

2 u u u w
Now
x 2 x x v w v w

2 2 2
u u 2 u u ...(iii)
x 2 v 2 v w w 2

v w
Again c and c
t t
u u u u w u u
. . c
t v t w t v w

2
u 2 u u
c
t 2 v w v w
2 u 2 u 2 u
2
c 2 2 2 ...(iv)
v v w w
Substituting from (iii) and (iv) in (i), we get
2 u 2 u 2 u
= c 2 2 2 2
v v w w

2 u 2 u 2 u
= c 2 2 2 2
v v w w

2
u
or 0
v w
Integrating with respect to w, we get

u
F v
v
where F(v) is an arbitrary function of v.
Integrating this with respect to v, we get

u v w .

where f v dv v

and (w) is an arbitrary function of w.
u , x t x ct x ct ...(v)
This is known as D, Alembert s Solution of the wave equation (i).

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