1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

446 Answers to Odd-Numbered Exercises

Finallyu(x)=(U−^3 /^2 +( 3 / 2 )

√

2 γ^2 / 5 x)−^2 /^3.

27. 459.77 rad/s.
    29.u(x)=C 0 e−ax.

31.w(x)= P

2 γ^2

[ 1

4

−x^2 +cosh(γx)−cosh(γ/^2 )

γsinh(γ/ 2 )

]

.

33. Thesolutionbreaksdown(bucklingoccurs)iftan(λ/ 2 )=γ/2.

Chapter 1

Section 1.1

1. a. 2

(

sin(x)−^1

2

sin( 2 x)+^1

3

sin( 3 x)−···

)

;

b.π

2

−^4

π

(

cos(x)+^1

9

cos( 3 x)+^1

25

cos( 5 x)+···

)

;

c.^1

2

+^2

π

(

sin(x)+^1

3

sin( 3 x)+^1

5

sin( 5 x)+···

)

;

d.^2

π

−^4

π

( 1

3

cos( 2 x)+^1

15

cos( 4 x)+^1

35

cos( 6 x)+···

)

.

3.f(x+p)= 1 =f(x)for anypand allx.

5. Ifcis a multiple ofp,thegraphoff(x)betweencandc+pis the same
   as that between 0 andp.Otherwise,letkbe the integer such thatkplies
   betweencandc+p:
   ∫c+p

c

f(x)dx=

∫kp

c

f(x)dx+

∫c+p

kp

f(x)dx=

∫p

c∗

f(x)dx+

∫c∗

0

f(x)dx,

wherec∗=c−(k− 1 )p.

7. a. cos^2 (x)=

1

2 +

1

2 cos(^2 x);

b. sin

(

x−π

6

)

=cos

(π

6

)

sin(x)−sin

(π

6

)

cos(x);

c. sin(x)cos( 2 x)=−^1

2

sin(x)+^1

2

sin( 3 x).

Section 1.2

1. a.

1

2 −

4

π^2

[

cos(πx)+

1

9 cos(^3 πx)+

1

25 cos(^5 πx)+···

]

;