1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

466 Answers to Odd-Numbered Exercises

and similarly

∂^2 u

∂t^2 =c

2

(∂ (^2) v
∂w^2 −^2
∂^2 v
∂z∂w+
∂^2 v
∂z^2

)

.

(We have assumed that the two mixed partials∂^2 v/∂z∂wand∂^2 v/∂w∂z

are equal.) Ifu(x,t)satisfies the wave equation, then

∂^2 u

∂x^2

=^1

c^2

∂^2 u

∂t^2

.

In terms of the functionvand the new independent variables this equa-

tion becomes

∂^2 v

∂w^2

+ 2 ∂

(^2) v
∂z∂w

+∂

(^2) v
∂z^2

=∂

(^2) v
∂w^2

− 2 ∂

(^2) v
∂z∂w

+∂

(^2) v
∂z^2
or, simply,
∂^2 v
∂z∂w

= 0.

13.u(x,t)=−c^2 cos(t)+φ(x−ct)+ψ(x+ct).

Section 3.4

1. Iffandgare sectionally smooth andfis continuous.


2. The frequency iscλnrads/sec, and the period is 2π/cλnsec.


3. Separation of variables leads to the following in place of Eqs. (11)
   and (12):
   T′′+γT′+λ^2 c^2 T= 0 , (11′)
   (
   s(x)φ′

)′

−q(x)φ+λ^2 p(x)φ= 0. (12′)

The solutions of Eq. (11′) all approach 0 ast→∞,ifγ>0.

7. The period ofTn(t)=ancos(λnct)+bnsin(λnct)is 2π/λnc.AllTn’s have
   a common periodpif and only if for eachnthere is an integermsuch
   thatm( 2 π/λnc)=p,orm=(pc/ 2 π)λnis an integer. Forλnas shown
   andβ=q/r,whereqandrare integers, this means

m=

(

pc

2 π

)

α

(

n+q

r

)

or

m=

(pc

2 π

)α

r(rn+q).