1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 3 467

Figure 4 Solution for Exercise 3, Section 3.6.

Givenα,pcan be adjusted so thatmis an integer whenevernis an integer.

Section 3.5

1. Ifq≥0, the numerator in Eq. (3) must also be greater than or equal to 0,
   sinceφ 1 (x)cannot be identically 0.


2. 2π^2 /3isoneestimatefromy=sin(πx).

5.

∫ 2

1

(y′)^2 dx=^1

3

,

∫ 2

1

y^2

x^4

dx=^25

6

−6ln2;

N(y)/D(y)= 42 .83;λ 1 ≤ 6 .54.

Section 3.6

1.u(x,t)=^12 [fe(x+ct)+Go(x+ct)]+^12 [fe(x−ct)−Go(x−ct)], wherefe

is the even extension offandGois the odd extension ofG.

3. See Fig. 4.


4. See Fig. 5.

7.u(x,t)=

1

2

[

f(x+ct)+f(x−ct)

]

+

1

2 c

∫x+ct

x−ct

g(y)dy.

Chapter 3 Miscellaneous Exercises

1.u(x,t)=

∑∞

1

bnsin(λnx)cos(λnct),bn= 2

(

1 −cos(nπ)

)

/nπ,λn=nπ/a.