THE LAPLACE TRANSFORM

8. THE LAPLACE TRANSFORM

As you will see in Section 9, Laplace transforms are useful in solving differential equations (for other uses see end of Section 9, page 442). Here we want to define the Laplace transform and obtain some needed formulas. We define L(f ), the Laplace transform of f (t) [also written F (p) since it is a function of p ], by the equation

L(f ) =

f (t)e −pt dt = F (p).

This is an example of an integral transform (also see Fourier transforms, Chapter 7, Section 12, and Hilbert transforms, Chapter 14, page 698). If we start with a function f (t), multiply by a function of t and p, and find a definite integral with respect to t, we have a function F (p) which is called an integral transform of f (t). There are many named integral transforms which you may discover in tables and computer. Observe the notation for Laplace transforms in (8.1): we shall consis- tently use a small letter for the function of t, and the corresponding capital letter for the transform which is a function of p, for example f (t) and F (p), or g(t) and G(p), etc. Also note from (8.1) that since we integrate from 0 to ∞, F (p) is the same no matter how f (t) is defined for negative t. However, it is desirable to define

f (t) = 0 for t < 0 (see footnote, page 447; also see Bromwich integral, page 696). It is very convenient to have a table of corresponding f (t) and F (p) when we are using Laplace transforms to solve problems. Let us calculate some of the entries in the table of Laplace transforms at the end of the chapter (pages 469 to 471). Note that numbers preceded by L (L1, L2, · · · , L35) refer to entries in the Laplace transform table.

Example 1. To obtain L1 in the table, we substitute f (t) = 1 into (8.1) and find

We have assumed p > 0 to make e −pt zero at the upper limit; if p is complex, as it may be, then the real part of p must be positive (Re p > 0), and this is the restriction we have stated in the table for L1.

Example 2. For L2, we have

e −(a+p)t dt =

Re(p + a) > 0.

0 p+a

We could continue in this way to obtain the function F (p) corresponding to each f (t) by using (8.1) and evaluating the integral. However, there are some easier methods which we now illustrate. First observe that the Laplace transform of a sum of two functions is the sum of their Laplace transforms; also the transform of

438 Ordinary Differential Equations Chapter 8

cf (t) is cL(f ) when c is a constant:

L[f (t) + g(t)] =

[f (t) + g(t)]e −pt dt

f (t)e −pt dt +

g(t)e −pt dt = L(f ) + L(g),

L[cf (t)] =

cf (t)e −pt dt = c

f (t)e −pt dt = cL(f ).

In mathematical language, we say that the Laplace transform is linear (or is a linear operator —see Chapter 3, Section 7).

Example 3. Now let us verify L3. In (8.3), replace the a by −ia; then we have

Re(p − ia) > 0.

p − ia

p +a

Remembering (8.4), we can write (8.5) as p

a (8.6)

L(cos at + i sin at) = L(cos at) + iL(sin at) = +i

. +a 2 p 2 +a 2

Similarly, replacing a by ia in (8.3), we get

(8.7) L(cos at − i sin at) =

−i

Re(p + ia) > 0.

p 2 +a 2 p 2 +a 2

Adding (8.6) and (8.7), we get L4; by subtracting, we get L3. Example 4. To verify L11, start with L4, namely

. Differentiate (8.8) with respect to the parameter a to get

L(cos at) =

p(−2a)

which is L11. Ways of finding other entries in the table are outlined in the problems.

PROBLEMS, SECTION 8

1. For integral k, verify L5 and L6 in the Laplace transform table. Hint: From L2, you R can write: ∞ 0 e −pt e −at dt = 1/(p + a). Differentiate this equation repeatedly with respect to p. (See Chapter 4, Section 12, Example 4, page 235.) Also note L32. For the Γ function results in L5 and L6, see Chapter 11, Problem 5.7.

2. By using L2, verify L7 and L8 in the Laplace transform table.

Section 8 The Laplace Transform 439

3. Using either L2, or L3 and L4, verify L9 and L10. 4. By differentiating the appropriate formula with respect to a, verify L12. 5. By integrating the appropriate formula with respect to a, verify L19. 6. By replacing a in L2 by a + ib and then by a − ib, and adding and subtracting the

results [as in (8.6) and (8.7)], verify L13 and L14. 7. Verify L15 to L18, by combining appropriate preceding formulas using (8.4).

Find the inverse transforms of the functions F (p) in Problems 8 to 13. 1+p

8. (p + 2) 2

Hint: Use L6 and L18. 9. 5 − 2p

Hint: Use L7 and L8. +p−2

10. 2p − 1 p 2 − 2p + 10

Hint: You can use L7 and L8 with complex a

and b, but L13 and L14 are more direct. 3p + 2

p 2 + 4p + 20 14. Show that a combination of entries L3 to L10, L13, L14 and L18 in the table, will

3p 2

p 2 − 25

give the inverse transform of any function of the form Ap + B

, where A, B, C, E, and F are constants. Cp 2 + Ep + F 15. Prove L32 for n = 1. Hint: Differentiate equation (8.1) with respect to p.

16. Use L32 and L3 to obtain L11.

17. Use L32 and L11 to obtain L(t 2 sin at).

18. Use L31 to derive L21. Table entries L28 and L29 are known as translation or shifting theorems. Do Problems

19 to 27 about them. 19. Prove the general formula L29 using (8.1). 20. Use L29 to verify L6, L13, L14, and L18. 21. Use L29 and L11 to obtain L(te − at sin bt) which is not in the table. 22. Obtain L(te − at cos bt) as in Problem 21. 23. Use the results which you have obtained in Problems 21 and 22 to find the inverse

transform of (p 2 + 2p − 1)/(p 2 + 4p + 5) 2 .

24. Sketch on the same axes graphs of sin t, sin(t − π/2), and sin(t + π/2), and observe which way the graph shifts. Hint: You can, of course, have your calculator or computer plot these for you, but it’s simpler and much more useful to do it in your head. Hint: What values of t make the sines equal to zero? For an even simpler example, sketch on the same axes y = t, y = t − π/2, y = t + π/2.

25. Use L28 to find the Laplace transform of

sin(t − π/2), t > π/2, 0,

f (t) =

t < π/2.

26. Use L28 and L4 to find the inverse transform of pe − pπ /(p 2 + 1). 27. Find the transform of

f (t) =

sin(x − vt), t > x/v, 0,

t < x/v,

where x and v are constants.

440 Ordinary Differential Equations Chapter 8