Differential Equations Exam Review (1 of 6) - Review 5...

1
Review 5: L29-L32
1. Solve the initial value problem by using the Laplace transform:
(a)
2 y y y t
′′ ′
− + =
,
( ) ( )
0 0, 0 1 y y
′
(b) cos2 w w t
′′+
=
0
2
w
π
 
=
 
 
,
1
2
w
π
 
′ =
 

(c) 3 y ty y t
′′ ′
+ − =
( ) ( )
0 0, 0 0 y y
′
2. (a) Sketch the graph of the piecewise continuous function and express it in terms of
the unit step function.
=

=
(b) Find L
( )
{ }
( )
f t s .
3. Find the following transforms:
(a) L 1
4
1
( 2)
s e
s
− −
 
 
−
 
(b) L 1
4 2
1
s e
s s
− −
 
 
+
 
(c) L 1
2 4 8
s
s
e
s s
− −
 
 
− +
 
(d) L 1
2
1
6 8
s e
s s
− −
 
 
− +
 
4. (a) Represent
( )
g t in terms of the unit step functions and determine L
( )
{ }
( )
0, 0 1
1, 1 2
0, 2
t
g t t
t
< <


= − < <


>

(b) Use part (a) to solve the initial value problem:
( )
4 y y g t ′′+ =
( ) ( )
0 0, 0 1 y y ′ =
.
5. (a) Represent
( )
g t in terms of the unit step functions and determine L
( )
{ }
g t
( )
, 0 1
2 , 1 2
0, 2
t t
g t t t
t
< <


= − < <


>

(b) Use part (a) to solve the initial value problem:
( )
4 y y g t
′′+
= ,
( ) ( )
0 1, 0 0 y y
′
=
4
3 2
g t
:
.