1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Chapter 6 485

Section 6.2

1. a.e^2 t;b.e−^2 t;c.^3 e

−t−e− 3 t

2 ;d.

sin( 3 t)

3.

3. a.^1 −e

−at

a

;b.t−sin(t);c.

sin(t)−^12 sin( 2 t)

3

;

d.(sin( 2 t)− 2 tcos( 2 t))/16; e.−

3

4 +

1

2 t+e

−t−^1

4 e

− 2 t;f.cosh(t)−1.

5. a.

e^2 t−e−^2 t

4 ;b.

1

2 sin(^2 t);

c.^3

2

+i

√

2 − 3

4

exp

(

−i

√

2 t

)

−i

√

2 − 3

4

exp

(

i

√

2 t

)

;d.4

(

1 −e−t

)

.

7. a. 1−cos(t);b.e

t−cos(ωt)+ωsin(ωt)

ω^2 + 1 ;c.t−sin(t).

Section 6.3

1. a.s=−

( 2 n− 1

2

π

) 2

,n= 1 , 2 ,...;

b.s=±i^2 n−^1

2

π,n= 1 , 2 ,...;

c.s=±inπ,n= 0 , 1 , 2 ,...;

d.s=iη,wheretanη=−^1

η

;

e.s=iη,wheretanη=^1

η

.

3. a.

sinh(√sx)

s^2 sinh(√s);b.

1

s−

cosh

(√

s(^12 −x)

)

s(s+ 1 )cosh(√s/ 2 ).

5. a.u(x,t)=x+

∑∞

n= 1

2sin(nπx)

nπcos(nπ)exp

(

−n^2 π^2 t

)

;

b.u(x,t)is 1 minus the solution of Example 3.

Section 6.4

1.t+x

2

2.

3.v(x,t)=^4

π^2

∑∞

1

cos

(

( 2 n− 1 )(^12 −x)

)

sin(( 2 n− 1 )πt)

( 2 n− 1 )^2 sin

( 2 n− 1

2 π

).