1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

480 Answers to Odd-Numbered Exercises

7. Taking the hint and using the fact that∇^2 φ=−λ^2 φ, the left-hand side
   becomes
   (
   λ^2 k−λ^2 m

)∫∫

R

φkφm

while the right-hand side is zero, because of the boundary condition.

Section 5.5

1.λn=αn/a,whereαnis thenth zero of the Bessel functionJ 0 .Thesolutions

areφn(r)=J 0 (λnr)or any constant multiple thereof.

3. This is just the chain rule.


4. Rolle’s theorem says that if a differentiable function is zero in two places,
   its derivative is zero somewhere between. From Exercise 4 it is clear that
   J 1 must be zero between consecutive zeros ofJ 0. Check Fig. 7 and Table 1.


5. Use the second formula of Exercise 6, after replacingμbyμ+1onboth
   sides.
   9.u(r)=T+(T 1 −T)I 0 (γr)/I 0 (γa).

Section 5.6

1.v( 0 ,t)/T 0 ∼= 1 .602 exp(− 5. 78 τ)− 1 .065 exp(− 30. 5 τ),whereτ=kt/a^2.

3.v(r,t)=

∑∞

n= 1

anJ 0 (λnr)exp(−λ^2 nkt),λn=αn/a.UseEq.(13)andothersto

findan=T 0 J 1 (αn/ 2 )/αnJ^21 (αn).

5. Integration leads to the equality
   ∫a

0

(

rφ′^2

)′

dr+λ^2

∫a

0

r^2

(

φ^2

)′

dr= 0.

The first integral is evaluated directly. The second must be integrated by

parts.

Section 5.7

1. Use
   1
   r

d

dr

(

rd

dr

J 0 (λr)

)

=−λ^2 J 0 (λr).

3. The frequencies of vibration areλmnc=αmnc/a.Thefivelowestvaluesof
   αmn, in order, have subscripts( 0 , 1 ),( 1 , 1 ),( 2 , 1 ),( 0 , 2 ),and( 3 , 1 ).See
   Table 1 in Section 5.6.