1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

6.3 Partial Differential Equations 381

Part a. (r 0 =0.) The limit assapproaches zero ofsU(x,s)may be found by
L’Hôpital’s rule or by using the Taylor series for sinh and cosh. From the latter,

sU(x,s)=

sx

(

1 +s 22 +···

)

−

(

sx+s^36 x^3 +···

)

s^2

(

1 +s 22 +···

)

=

s^3

(x

2 −

x^3

6 −···

)

s^2

(

1 +s 22 +···

)→ 0.

Thus, in spite of the formidable appearance ofs^3 in the denominator,s=0is
not really a significant value and contributes nothing tou(x,t).

Part b. It is convenient to take the remaining roots in pairs. We label

±i( 2 n− 1 )π

2 =±iρn.

The derivative of the denominator is

p′(s)= 3 s^2 cosh(s)+s^3 sinh(s),

p′(±iρn)=±i^3 ρ^3 nsinh(±iρn)

=ρ^3 nsin(ρn)

since sinh(iρ)=isin(ρ)and(±i)^4 =1. The contribution of these two roots
together may be calculated using the exponential definition of sine:

q(iρn)

p′(iρn)

exp(iρnt)+q(−iρn)

r′(−iρn)

exp(−iρnt)

=−sinh(iρnx)exp(iρnt)+sinh(iρnx)exp(−iρnt)

ρn^3 sin(ρn)

=ρsin 3 (ρnx)

nsin(ρn)

i

(

−exp(iρnt)+exp(−iρnt)

)

=

2sin(ρnx)sin(ρnt)

ρ^3 nsin(ρn).

Part c. The final form ofu(x,t), found by adding up all the contributions
from Part b, is the same as would be found by separation of variables

u(x,t)= 2

∑∞

1

sin(ρnx)sin(ρnt)

ρ^3 nsin(ρn).