# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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(s - 2) + 1

where F(s) = (s2 + 1)-1. Since L-1{F(s)} = sin t, it follows from Theorem 6.3.2 that

g(t) = C~\G (s)} = e2t sin t.

The results of this section are often useful in solving differential equations, particularly those having discontinuous forcing functions. The next section is devoted to examples illustrating this fact.

In each of Problems 1 through 6 sketch the graph of the given function on the interval t > 0.

2. (t - 3)u() - (t - 2)è()

4. f (t - 3)u3(t), where f (t) = sin t

1. u1(t) + 2u3(t) - 6u4(t)

3. f (t - n)un(t), where f (t) = t2

5. f (t - \)u2(t), where f (t) = 2t

6. f (t) = (t - 1)uj(t) - 2(t - 2)u2(t) + (t - 3)u3(t)

In each of Problems 7 through 12 find the Laplace transform of the given function.

7. f(t) = ¦ 0, 2)2, t < 2 8. f (t) = 0, 2t - 2t + 2, t < 1

(t - t>2 t > 1

0, t < n

9. f(t) = ¦ t- n, n < t < 2n 10. f (t) = -- u1(t) -) 6u4(t)

(

è

2

+

0, t > 2n

1. (f -) t 12. f (t) = -- t - è (t)(t - 1), t > 0

) (2 1

= Ç )2

t uF

1 )t

)3

In each of Problems 13 through 18 find the inverse Laplace transform of the given function.

e-2s

13. F (s) = 3! 14. F (s)

(s - 2)4

15. F (s) = 2(s - 1)e-2s 16. F (s)

s2 --- 2s + 2

17. F (s) = (s - 2)e-s 18. F(s)

s2 - 4s + 3

s2 + s — 2

2e

2s

s2 4

e-s + e-2s - e-3s - e-4s

6.3 Step Functions

315

19. Suppose that F(s) = L{ f (f)} exists for s > a > 0.

(a) Show that if c is a positive constant, then

c \c

(b) Show that if k is a positive constant, then

L{ f (ct)} = , s >

constant, then

^ f<ks)}=¯ '(0-

kk

(c) Show that if a and b are constants with a > 0, then

L-1{F(as + b)}= 1 e-bt/af(^j .

In each of Problems 20 through 23 use the results of Problem 19 to find the inverse Laplace transform of the given function.

20. F(s) =

2ë+'ë!

ñë+1

22. F(s) =

9s2 - 12s + 3

2s + 1

21. F (s) = —2---------------------

4s2 + 4s + 5

23. F (s) =

2s- 1

In each of Problems 24 through 27 find the Laplace transform of the given function. In Problem 27 assume that term-by-term integration of the infinite series is permissible.

24. f(t) =

0 < t < 1 t> 1

25. f(t) =

0 < t < 1 1 < t < 2 2 < t < 3 t3

2ë+1

26. f (t) = 1 - u1(t) + •••+ u2n (t) - u2n+1(t) = 1 + ? (-1)kuk(t)

k=1

TO

27. f (t) = 1 + (-1)kuk(t)- See Figure 6.3.6.

1

e

FIGURE 6.3.6 A square wave.

28. Let f satisfy f (t + T) = f (t) for all t > 0 and for some fixed positive number T; f is said to be periodic with period T on 0 < t < to. Show that

[ e-st f(t) dt L{ f (0} =Jo

1 -

316

Chapter 6. The Laplace Transform

In each of Problems 29 through 32, use the result of Problem 28 to find the Laplace transform of the given function.

1, 0 < t < 1,

29. f(t) =

0, 1 < t < 2;

f (t + 2) = f (t).

Compare with Problem 27.

30. f(t) =

1,

— 1,

f (t + 2) = f (t). See Figure 6.3.7.

0 < t < 1, 1 < t < 2;

31. f(t) = t, 0 < t < 1;

f (t + 1) = f (t).

See Figure 6.3.8.

32. f (t) = sin t, 0 < t < n;

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