Engineering Mathematics  Exam III
By Shun-Feng Su
Jan. 4, 2014
一. Short Answers. 5 points each.
(1) £((t+2) 2 +t+1) =
.
(2) £(((t+2) 2 +t+1) H (t  1) ) =
.
(3) £(((t+2) 2 +t+1) H (t  1) ) =
.
(4) £((2－t) H (2  t ) ) =
.
(5) £(t 2 cos(t  1) ) =
.
(6) £( sin(t ) H (t 2  2t  3) ) =
(7) £(
t (n) t
f ( )) =
a
a
.
(in terms of F(s)= £( f (t ) ); a is a constant).
(8) £(e at cos(t  a) H (t  a) ) =
.
(9) £( t (t 2  3t  2) )=
(10) £ 1 (2 e 3s )=
.
.
2
e 3 s 1 1
(11) £ ( 2 (  2  3 ) ) =
s 2s
3s
s
1
(12) £ 1 (
s( s  1)  2
)=
s( s  1)2 ( s  2)
.
.
(13) £ 1 (
s( s  2)
)=
s  4s  13
.
2
a2
(14) £ ( ln(1  2 ) ) =
s
1
.
(15) Express the function shown in the following figure in terms of the
Heaviside functions.
(16) Find the Laplace transform of the above function.
3
(17) Evaluate
cos(t )
 ( t  2 )  (t   )dt .
2
0
(18) Find the convolution of (t1) e 2 t and  (t  3) .
(19) Please give a function whose Laplace transform does not exist.
(20) Suppose that both f(t) and g(t) are piecewise continuous on [0, k]
for every k >0 and also both are O(e bt ) . If their Laplace transforms
are equal, then under what situations f(t)  g(t)?
(21) Suppose that f(t) is continuous for t >0 and is O(e bt ) . Let
1
£( f (t ) )=F(s) and £(g(t))= F(s). Then write g(t) in terms of f(t).
s
(22) Suppose that f(t) is piecewise continuous for t0 and is O(e bt ) .
1
Let £( f (t ) )=F(s). Then, write £( f(t)) in term of F(s).
t
二. True/False. Be sure to justify your answers. 5 points each.
(1) Let g(t) and h(t) both be piecewise continuous for t0. If
| h(t ) | g (t ) for some interval [0, a] and if

 g (t )dt
converges,
a

then
 h(t )dt
also converges too.
0
(2) Since a square wave function has an infinite number of
discontinuities, it is not a piecewise continuous function. However,
its Laplace transform still exists.
(3) If lim £( f (t ) )(s) = 0, then the Laplace transform of f(t) exists.
s
(4) Suppose that the Laplace transform of f(t) exists as F(s). Let a > 0.
t
1
1 a
Then £(  f ( )d )= F ( s )   f ( )d .
a
s
s 0
(5) H (a  t )  H (t )  H (t  a) for all t0.
(6) If £( f (t ) )=F(s) exists for s>b, then no matter what the constant a is,
we can always claim that £( e at f (t ) )=F(s-a) for s>a+b.
(7) If the convolution of f(t) and g(t) exists, then both the Laplace
transforms of f(t) and g(t) also exist.
(8) Suppose that f (t ) , g 1 (t ) , and g 2 (t ) are all continuous. If
f  g1 (t ) = f  g 2 (t ) for t > 0, then necessarily g 1 (t ) = g 2 (t ) for all
t > 0.
(9) If £( f1 (t ) )=£( f 2 (t ) ) and f1 (t ) = f 2 (t ) , then f1 (t ) and f 2 (t )
must be both continuous.
(10) Suppose that f(t) is integrable for t >0. Let a>0. Then

 f (t ) (t  a)dt  f (a) .
0
三. (10 points) Find the solution for y   2 y   f (t ) ; y(1)=1, y (1) =0;
where f(t)=0 when t 0; f(t)=2 when 1  t < 3; and f(t)=2t3 when t  3.
四. (20 points) Find the Laplace transform of the half-wave rectification of
 sin(t ) as shown in the following.
Problem 六
五. (10 points) Suppose that F(s) is the Laplace transform of f(t). Please
show that £(t f (t ) )= 
dF ( s )
, where.
ds
六. (10 points) If f  g is defined, show that f  g = g  f .
七. (20 points) Suppose that f(t), f (t ) , …, f
for t > 0 and that f
(n)
( n1)
(t ) are all continuous
(t ) is piecewise continuous on [0, k] for every
k >0. Suppose also that each of f(t), f (t ) , …, f
( n1)
(t ) is O(e bt ) .
Please show by induction that for n= 1, 2, …, n, …,
£( f
(n)
(t ) )= s n £( f (t ) )  s n 1 f(0) s n  2 f (0) …  f
( n 1)
(0) .
t
八. (10 points) Find f(t) in e3t f (t )  et   3 f ( )e3 d .
0
九. (15 points) Find the Laplace transform of
十.
1
(1  cosh(at )) .
t
(20 points) Solve (1  t ) y  ty  y  0 ; y(0)=3, y(0)  1 .
十一. (20points) Solve (t  1)(2  t ) y  2 y  2 y  6t  6 ; y(1)  0 .
十二. (15 points) Solve the following differential equation system
x  2 y  y  1 ;
2 x  y  0 ;
The total score is 310. Do your best and good luck.