Differential Equations
for Engineers:
the Essentials
Class 14 notes
Agenda: Class 14
Working on a qualitative aspect of ODEs: system stability
Lectures:
Questions on Determining System Stability
What do the following tell you with respect to system stability?
(a) negative coefficients in any linear time-invariant ODE
Introduction to Laplace Transforms
Value of Laplace Transforms
For linear time-invariant ODEs of any order,
Laplace transforms:
convert differential equations into algebraic ones
solve for initial conditions without extra steps
Preliminaries
 
→
=
a
A
a
Adttfdttf )(lim)(
Improper integral:
t
Piecewise continuous:
A function is piecewise continuous on if it is
Laplace Transform
Definition:
The Laplace transform of the function is
Existence:
 
−
== 0)()()( dttfetfLsF st
)(sF
)(tf
Laplace Transforms of
Some Common Functions
as
at
e
Laplace Transforms of
Some Common Functions (2)
t
sin
 
 
( )
 
−−
→
−
→
−
+−==
==
0
0 0
cossin
1
limsin
sinlimsinsin
dtte
s
Te
s
tL
tdtetdtetL
TstsT
T
Tst
T
st
Integrating by parts twice,
Key Laplace Transform Attribute
Theorem:
Let be continuous and of exponential order and be
piecewise continuous on any interval
Then exists and
)(tf
dtdf /
At 0
 
dtdfL/
Key Laplace transform attribute (2)
Corollary:
Let be continuous and of exponential order
and let be piecewise continuous on any interval .
1
1
2
2
,…,,, −
−
n
n
dt
fd
dt
fd
dt
df
f
n
n
fd
At 0
More Laplace Transforms of
Common Functions
t
cos
 
 
)0(sinsin
1
)(sin
1
cos
=−=
=
tsLt
dt
d
LtL
Recall from before that
 
22
sin
+
=s
tL
Then
Short Table of Laplace Transforms
2
1
)(
1
)(
+
s
s
sF
)(tf
t
e
−
t
Laplace transforms for solving
initial value problems
Using Laplace Transforms to Solve
Ordinary Differential Equations
Steps to solving linear time-invariant ordinary differential equations
(homogeneous or nonhomogeneous):
(1) Transform each element of the given ODE from a function of time into
an algebraic function of s.
Laplace Transform Example #1
02 =+ y
dt
dy
5)0( =y
Problem: Solve the ODE
Laplace Transform Example #1 (2)
 
5)0( −=−=
sYyysL
dt
dy
L
Laplace transform of a derivative:
From the definition of a Laplace transform:
Equation 4
Laplace Transform Example #1 (3)
Equation 7 is called the solution in the Laplace Transform
domain.
Laplace Transform Example #2
0134
2
2
=++ y
dt
dy
dt
yd
Problem: Solve
4)0( =y
1)0( −=y
Solution:
 
0134
2
2
Ly
dt
dy
dt
yd
L
=
++
Equation 8
Laplace Transform Example #2 (2)
Now define
 
)()( sYtyL =
Then
4)0(
−=−=
sYysY
dt
dy
L
Equation 11
Equation 10