Differential Equations with Piecewise Continuous Functions and Periodic Functions
Definition 1: The unit step function u (t ) is defined by
0, t < 0
.
u (t ) = 
1, t ≥ 0
Suppose a ≥ 0 , what’s u (t − a ) ?
0, t < a
u (t − a ) =

1, t ≥ a
We still use ua (t ) to denote u (t − a ) , or ua (=
t ) u (t − a )
 1, a ≤ t < b
Also, we use notation uab (t ) = u (t − a ) − u (t − b) = 
0, t < a, t ≥ b
What’s Laplace transform of u (t − a ) ?
 3, t < 2
 1, 2 < t < 5

Example: Write the following function f (t ) = 
in terms of unit step
t
,5
<
t
<
8

t 2 /10,8 < t
functions.
Property 1: Let F ( s ) = L( f )( s ) , and a ≥ 0 . We have
L( f (t − a )u (t − a ))( s ) =
e − as F ( s ) , and
L−1 (e − as F ( s ))(t ) =f (t − a )u (t − a ) .
Property 2: L( g (t )u (t − a ))( s=
) e − as L( g (t + a ))( s ) .
Example 2: Determine the Laplace transform of t 2u (t − 1) .
 e −2 s 
Example : Determine L  2  .
 s 
−1
 e −3 s 
Example : Determine L  2
.
 s +s−2
−1
Looking at the following differential equations coming from the modeling of on/off
switches, changes. The current I in an LC series circuit is governed by the initial value
problem:
(1)
I ''(t ) + 4 I (t )= g (t ), I (0)= 0, I '(0)= 0
where
 1, 0 < t < 1,

g (t ) = −1,1 < t < 2
 0, 2 < +∞

Example : Solve the above initial value problem.
Definition 2: A function f (t ) is said to be periodic of period of T if f (=
t ) f (t + T ) for
all t in the domain of f.
Property 3: If f has period T and is piecewise continuous on [0,T], then
∫
L( f )( s ) =
T
0
e − st f (t )dt
1 − e − sT
e t , 0 < t < 1
Example : Determine L( f )( s ) , where f (t ) = 
and f (t ) has a period 2.
 0,1 < t < 2
Example : Find the inverse Laplace transform of
F (s) =
1 − e− s
s (1 − e −2 s )
Example: Solve the initial value problem
 y ''+ π 2 y =
f (t ),

 y (0) = 0
 y '(0) = 0

1, 0 ≤ t < 1,
where f (t ) = 
and f (t ) has period 2.
0,1 ≤ t < 2,