Course: Engineering Mathematics I (MATH 1180)
Assignment #5: LAPLACE TRANSFORMS
Group I: CIV, ELEC, SURV.
MATH 1180:  Assignment #6: Laplace Transforms
Lecturer: Mr. Oral Robertson
Date: November 26, 2010
Due on: Tuesday, November 30, 2010.
1. Proofs:
(a) Prove from first principles that: L[1] =
1
and L[eat f(t)] = F(s – a).
s
(b) Starting with the definitions of the hyperbolic functions sinh at and cosh at, show that:
L[sinh at ] 
a
s
.
and L[cosh at ]  2
2
s a
s  a2
2
 F ( s) 
(c) Use the Convolution Theorem to show that: L 

 s 
1
(d) Show, from first principles that: L[ H (t  a)] 
f
 f (t ) dt
.
0
e  as
.
s
2. Piecewise-continuous functions and the Heaviside step-function:
The Heaviside step-function H(t – a), which is really a multiplier taking a value of zero or one, and its
reverse-Heaviside step-function, 1 – H(t – a), are defined by:
t  a

t  a
0,
H (t  a)  
1,
1,
1  H (t  a)  
0,
and
t  a
.
ta
By expressing the following piecewise-continuous functions using Heaviside notation (assume t  0),
find their Laplace Transforms:  Recall: L[H(t – a).f(t – a)] = e–as F(s) .
 0 ,
(a) f t   
2
t  4  ,
t  4

t  4
t , t  4 
(b) f t   

0, t  4
cos t , t   
(c) f t   

sin t , t   
3. Functions active in an interval {a,b} :
It can be deduced that if functions f1(t), f2(t), and f3(t) are active only in the intervals t < a, a  t < b, and
t  b, respectively, then these functions are (independently) represented using Heaviside notation as:
[1 – H(t – a)]f1(t), [H(t – a) – H(t – b)] f2(t), and [H(t – a)] f3(t), respectively.
Use the above reasoning and/or a combination of the Heaviside and reverse-Heaviside step-functions (as
necessary) to find the Laplace transform of each of the following functions:
1, t  2

(a) f t   0, 2  t 
1, t  4



4


t 1 
t ,


(b) f t   2t , 1  t  3
3t , t  3 


t 
 1,



(c) f t   cos t ,   t  2 
 1,

t  2


4. Second Shift Theorem (Heaviside step-function):
(a) Find the Laplace Transform of the following:
(i) H (t   ). cos(t   )
(v) H t  1
. 2t  4
(ii) H t  1
. t  4
(vi) H t  1.e 2t
(iii) H t   . sin t
(b) Find the inverse Laplace Transform of the following:
e 2 s
e 2 s
e 2 s
(i ) 3
(ii) 2
(iii) 2
s
s 9
s 1
. t  2
(iv) H t  2
(iv) se  s ln s
(v)
2
se s
s2  4
5. Finding inverse L.T.’s by successive differentiation of F(s):
n
n d F ( s)
By using the theorem L [t n . f (t )]   1
to do successive differentiation of the following F(s)
ds n
functions (until they reach simple table format), find the inverse Laplace Transform of each:
(i) ln s

(ii) s ln s
(vii) ln s 2  a 2

(iii) s 2 ln s
(iv) tan 1 s
(viii) ln [s  a   b 2 ]
2
(v) s tan 1 s
sa
(ix) ln 

sa
(vi) ln( s  a)
a

(x) ln 1   .
s

1
Course: Engineering Mathematics I (MATH 1180)
Assignment #5: LAPLACE TRANSFORMS
Group I: CIV, ELEC, SURV.
Lecturer: Mr. Oral Robertson
Date: November 26, 2010
6. General Laplace Transform evaluation:
Find, using any appropriate method, the Laplace Transform of the following:
*(You may also apply any standard trigonometric identities to aid in your evaluation, if necessary.)
(i) e 3t sin 2t
(ii) e 3t cosh 3t
(vii) sin 2 t (viii) sin t cos t
(iii) t 2 e 3t
(vi) cos 2 t
(v) t 2 cos t
(iv) t sin t
(x) t e 2t sin t
(ix) t sin t cos t
7. General inverses (using 1st Shift Theorem and /or Partial Fractions, etc.)
(a) Find the inverse Laplace Transform of each of the following:
(i)
(vi)
1
s4
(ii)
1
s  2
3
1
s  1s  2
(iii)
s2
s  2
9
2s  4
(vii) 2
s s  1s 2  4 
2
(iv)
1
(v)
s  1
4
1
(viii)
2s  53
2
(ix)
1
s  8s  9
2
1
2s  5
3
(x)
s
s  13
(b) Show that if we transpose the two ‘versions’ of the 1st Shift Theorem for f(t), we get:
f(t) = eat F(s + a) and f(t) = e –at F(s – a).
By conveniently shifting the denominator of the given F(s) items to simply “s”, use the appropriate
one of the two results to find their inverse Laplace transform. Remember, all “s” terms must be
‘shifted’ anywhere they occur in the given F(s) function.
(i)
s
s  1
(ii)
s2
s  23
(iii)
s 2  2s  9
s  23
(iv)
s 3  2s 2
s  15
(v)
6 s  7 2
3s  4 3

 1 
 f (t ) 
1
(c) Use the Laplace Transform identity L 
to
verify
that:

F
(
s
)
ds
L
 te t .

2


 t  s
 s  1 
 sin t 
(d) Use the identity in part (c ) above to evaluate to find L 
 .
 t 
8. Multiple methods of calculating a particular inverse L.T. :
1
1
F (s)
(a) By treating
as
, use the corresponding L.T. identity to find: L1
.
2
2
s
s ( s  1)
s s  1
F (s)
1
1
(b) By further treating 2 2
as
, use your answer in part(a) to find: L1 2 2
.
s
s s  1
s ( s  1)
(c) Find L1
(d) Find L1
1
s ( s  1)
2
1
s ( s  1)
2
and L1
1
using partial fractions.
s ( s 2  1)
and L1
1
using the Convolution Theorem. {Treat each as F(s) G(s)}.
s ( s 2  1)
2
2
9. Use the Convolution Theorem to find: (i) L1
1
1
,
2
( s  1) 2
(ii) L1
in terms of partial fractions, show that L1
s2
s
, and (iii) L1 2
.
2
2
( s  1) 2
( s  1)
1
= 1 – cos t .
s ( s  1)
s ( s  1)
Use this result and other known L.T. identities to find the inverse Laplace Transform of the following:

1
d  1
e s
1
(i)
(ii)
(iii)
(iv) s s 2


2
2
2
( s  1) [( s  1)  1]
ds  s( s  1) 
s ( s  1)
3 9 1
10. By writing
2
2


11. Use Laplace Transforms to solve the following Differential Equations:
(i) y   3 y  13 sin 2t ; y (0)  6.
(ii) y   3 y   2 y  e 4t ; y(0)  6, y (0)  5.
(iii) y   y  e 3t cos 2t ; y(0)  0.
(iv) y   3 y   2 y  t e t ; y(0)  0, y (0)  0.
(v) y   2 y   5 y  0 ; y (0)  1, y (0)  3.
(vi) y   8 y   25 y  100 ; y (0)  2, y (0)  20.
2