1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

3.2 Solution of the Vibrating String Problem 219

∂^2 u

∂x^2

=^1

c^2

∂^2 u

∂t^2

, 0 <x<a, 0 <t, (1)

u( 0 ,t)= 0 , u(a,t)= 0 , 0 <t, (2)

u(x, 0 )=f(x), 0 <x<a, (3)

∂u

∂t(x,^0 )=g(x),^0 <x<a, (4)

contains a linear, homogeneous partial differential equation and linear, homo-
geneous boundary conditions. Thus we may apply the method of separation of
variables with hope of success. If we assume that^1 u(x,t)=φ(x)T(t), Eq. (1)
becomes

φ′′(x)T(t)=

1

c^2 φ(x)T

′′(t).

Dividing through byφT,weobtain

φ′′(x)

φ(x)=

T′′(t)

c^2 T(t),^0 <x<a,^0 <t.

For the equality to hold, both members of this equation must be constant.
We write the constant as−λ^2 and separate the preceding equation into two
ordinary differential equations linked by the common parameterλ^2 :

T′′+λ^2 c^2 T= 0 , 0 <t, (5)

φ′′+λ^2 φ= 0 , 0 <x<a. (6)

The boundary conditions become

φ( 0 )T(t)= 0 ,φ(a)T(t)= 0 , 0 <t

and, sinceT(t)≡0 gives a trivial solution foru(x,t),wemusthave

φ( 0 )= 0 ,φ(a)= 0. (7)
TheeigenvalueproblemEqs.(6)and(7)isexactlythesameastheonewe
have seen and solved before. (See Chapter 2, Section 3.) We know that the
eigenvalues and eigenfunctions are

λ^2 n=

(

nπ

a

) 2

,φn(x)=sin(λnx), n= 1 , 2 , 3 ,....

Equation (5) is also of a familiar type, and its solution is known to be

Tn(t)=ancos(λnct)+bnsin(λnct),

(^1) Tno longer symbolizes tension.