Assignment III
1. State and prove Convolution Theorem
2. Evaluate
 1 
L1 
4
 s  3 
I.
II. Le sin at
at
 e 2 s  e 3 s 
L1 

s

III. 
 e  bs 
L1  2
2
s   
IV.
  s  1 
L1 log

V.   s  1 

VI.
L t e  t cosh 2t
VII.
 e 4 s 
L  3 
 s 

1
3. Find Laplace transform of sin 2t u t    .
  s 2  4 s  5 
.
L log 2
4. Find   s  2s  5 
1


16
L1 
.
2
5. Use Convolution theorem to find  s  2s  2 
6. Using Laplace transform solve:
y  2 y  y  te  t under the conditions y 0   1, y0   2.
t t
  sin u dudu.
7. Find the Laplace transform of 0 0
1,
2,

F t   
3,
0,
8.
0t 2
2t 4
4t 6
t 6
9. Sketch the following functions and express them in terms of unit step
functions. Hence obtain the Laplace transform.
t 2
F t   
4t
0t 2
t2
y 0   0, y0   1.
d2y
dy
t  y  0
2
10. Solve dt
if
dt