Differential Equations
for Engineers:
the Essentials
Class 18 notes
Agenda: Class 18
Review of Short Quiz #2
Review of Homework Assignments 14 and 16
Lectures:
Complex Eigenvalues and Eigenvectors
Homework Assignment 21
Objective
In the previous class we examined an example system that
Systems of First Order ODEs
Electrical Circuit Example
Resistor
Rohms
Electrical Circuit Example (2)
R
1
i
R
1
i
Electrical Circuit Equations
From Kirchhoff’s laws:
Sum of voltage drops around loop 1 = 0
L
R
R
C
1
i
2
i
)(tV
Electrical Circuit Equations (2)
0)/)(2/1( 12 =−= RVii
Eq’n 4
From Equation 2
Using Equation 4 in Equations 1 and 3
Electrical Circuit Equations (3)
L
R
R
C
1
i
)(tV
To put this into state space format, let
=
1
i
V
x
Then Equations 5 and 6 become
Electrical Circuit Equations (4)
R
1
i
Let’s assume that
4
=
R
Electrical Circuit Equations (5)
To solve Equation 8 we try (as before)
rt
ex
=
Equation 9
Electrical Circuit Equations (6)
As always, Equation 10 results in two conditions. The first is
0)det( =− IrA
Only certain values of in our trial solution (Equation 9) will work.
Those values satisfy Equation 11. They are called eigenvalues.
r
Equation 11
Electrical Circuit Equations (7)
Substituting Equation 7 into Equation 11:
rr
r
r
IrA
8)2/1()2)(2(
22/1
82
det)det(
=−−−−−−
=
−−−
−−
=−
Electrical Circuit Equations (8)
The next step is to substitute and Equation 7 (for )
into Equation 12. The result is
ir 22
1+−=
A
+−−−
=−
0
8)22(2
0)(
)1(
1
)1(
1
i
IrA
Electrical Circuit Equations (9)
Writing out Equation 13 component by component we have
These represent just one equation (as they must, since the
determinant is zero). That equation can be written as
082
)1(
2
)1(
1
=+−
i
04 )1(
2
)1(
1=+−
i
Electrical Circuit Equations (10)
Hence the first eigenvector is given by
We could go through the whole procedure again for the second
but we need not. We will always find that when two eigenvalues
are complex conjugates then the eigenvectors will be complex
conjugates also. Hence we can immediately write
=i)4/1(
1
)1(
Electrical Circuit Equations (11)
The fundamental solutions associated with complex eigenvalues
can always be written
where and are vectors of real-valued constants.
a
b
ir +−=
1
ir −−=
2
and
Electrical Circuit Equations (12)
What do Equations 14 mean in real terms? As we did for second order
linear time-invariant equations with characteristic polynomial
having complex roots, we invoke Euler’s equation
We find that
tite ti
sincos +=