1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

368 Chapter 6 Laplace Transform

b. f(t)=

{ 0 , 0 <t<a,

1 , a<t<b,

0 , b<t;

c. f(t)=

{t, 0 <t<a,

a, a<t.

4.TheHeaviside step functionis defined by the formula

Ha(t)=

{ 1 , t>a,

0 , t<a.

Assuminga≥0, show that the Laplace transform ofHais

L

(

Ha(t)

)

=

e−as

s.

5.Use completion of square and the shifting theorem to find the inverse trans-

form of

a. s (^2) +^12 s, b. s (^2) +s+ 2 s^1 + 2 , c. s (^2) + 21 as+b 2 , b>a.
6.Find the Laplace transform of the square-wave function
f(t)=
{ 1 , 0 <x<a,
0 , a<x< 2 a, f(x+^2 a)=f(x).
Hint: Break up the integral as shown in the following, evaluate the integrals,
and add up a geometric series:
F(s)=

∑∞

n= 0

∫ 2 (n+ 1 )a

2 na

f(t)e−stdt.

7.Use any method to find the inverse transform of the following.

a. (s−a)(^1 s−b);

c. s

2

(s^2 +ω^2 )^2 ;

b.

s

(s^2 −a^2 )^2 ;

d. (s−^1 a) 3 ; e.^1 −e

−s

s

.

8.Use any theorem or formula to find the transform of the following.

a.

1 −cos(ωt)

t ;

c. t^2 e−at;

b.

∫t

0

sin(at′)

t′ dt

′;

d.tcos(ωt); e.sinh(at)sin(ωt).

9.Find the inverse transform of these functions ofsby any method.