Mathematical Physics Vol 1

7.6 The variable separation method

367

√

√

− λ · x

− λ · x

+ C 2 e −

X ( x )= C 1 e

(7.188)

,

X ( 0 )= C 1 + C 2 = 0 , X ( l )= C 1 e α

α

+ C 2 e −

= 0 ,

( α = l p − λ ) ⇒ α − e − α C 1 = C 2 = 0 ⇒ X ( x ) ≡ 0 .

C 1 = − C 2 i C 1 e

= 0 ⇒

(7.189) Thus, in this case, we have only the trivial solution. As we are not interested in this solution, we will now observe the remaining cases. 2 ◦ λ = 0

X ( x )= C 1 x + C 2 ,

(7.190)

X ( 0 )=( C 1 x + C 2 ) | x = 0 = C 2 = 0 , X ( l )= C 1 l = 0 ⇒ C 1 = C 2 = 0 ⇒ X ( x ) ≡ 0 .

(7.191)

Thus, in this case, we also have only the trivial solution. 3 ◦ λ > 0

X ( x )= C 1 cos √ λ x + C 2 sin √ λ x ,

(7.192)

X ( 0 )= C 1 = 0 , X ( l )= C 2 sin √ λ l = 0

(7.193)

⇒

if C 2̸ = 0 (nontrivial solution), then the following must be true sin √ λ l = 0 ⇒ √ λ = π n l = p λ n , and the nontrivial solution has the form For T ( t ) we now obtain for λ = λ n = π n l 2 T n ( t )= A n cos π n l Thus, the nontrivial solution of our problem is the function u n ( x , t )= X n ( x ) · T n ( t )= at + B n sin π n l at . X n ( x )= C n · sin π n l x .

(7.194)

(7.195)

(7.196)

(7.197)

= h A n cos

at + B n sin

at i sin

π n l

π n l

π n l

x .

According to the superposition principle, a solution is also the function

∞ ∑ n = 1

u =

u n ( x , t )=

(7.198)

∞ ∑ n = 1 h

A n cos

at + B n sin

at i sin

π n l

π n l

π n l

x .

=