Page 99 - Advanced engineering mathematics

3.1 Definition and Notation 79

to open the integral transforms package of subroutines. For the Laplace transform of f (t), enter

laplace(f(t),t,s);
to obtain F(s).
The Laplace transform is linear, which means that the transform of a sum is the sum of the
transforms and that constants factor through the transform:
L[ f + g]= F + G and L[cf ]= cF
for all s such that F(s) and G(s) are both defined and for any number c.

Given F(s), we sometimes need to find f (t) such that L[ f ]= F. This is the reverse process
of computing the transform of f , and we refer to it as taking an inverse Laplace transform.
−1
This is denoted L , and
−1
L [F]= f exactly when L[ f ]= F.

at
For example, the inverse Laplace transform of 1/(s − a) is e .
If we use Table 3.1 to find an inverse transform, read from the right column to the left
column. For example, using the table and the linearity of the Laplace transform, we can read that

3 1 3
−1 5t 12t
L − 7 = sin(4t) + e − e .
2
s + 16 (s − 5)(s − 12) 4
−1
The inverse Laplace transform L is linear because L is. This means that
−1
−1
−1
L [F + G]= L [F]+ L [G]= f + g,
and for any number c,
−1
−1
L [cF]= cL [F]= cf.
To use MAPLE to compute the inverse Laplace transform of F(s), enter
invlaplace(F(s),s,t);
to obtain f (t). This assumes that the integral transforms package has been opened.

SECTION 3.1 PROBLEMS

In each of Problems 1 through 5, use Table 3.1 to determine 7. Q(s) = s
s 2 +64
the Laplace transform of the function. 5 4s
8. G(s) = −
s 2 +12 s 2 +8
1. f (t) = 3t cos(2t) 1 1
9. P(s) = s+42 − (s+3) 4
2. g(t) = e −4t sin(8t)
−5s
10. F(s) = (s 2 +1) 2
3. h(t) = 14t − sin(7t)
For Problems 11 through 14, suppose that f (t) is defined
4. w(t) = cos(3t) − cos(7t)
for all t ≥0 and has a period T . This means that f (t + T )=
2 −4t
5. k(t) =−5t e + sin(3t) f (t) for all t ≥ 0.
In each of Problems 6 through 10, use Table 3.1 to 11. Show that
determine the inverse Laplace transform of the function.
∞ (n+1)T

L[ f ](s) = e −st f (t)dt.
7
6. R(s) = nT
s 2 −9 n=0
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October 14, 2010 14:14 THM/NEIL Page-79 27410_03_ch03_p77-120