Differential Equations
for Engineers:
the Essentials
Class 17 notes
Agenda: Class 17
Review Homework Assignment 15
Lectures:
Systems of First Order ODEs:
Homework Assignment 17
Short Quiz #2
Systems of First Order ODEs
Linear Time Invariant Example:
10
T
20
T
Heat Transfer Example
Two thin wafers of identical
size and material are separately
heated or cooled to initial
temperatures and .
10
T
20
T
Heat Transfer Example (2)
)()(
)()(
212
2
211
1
TT
d
kA
TThA
dt
dT
mc
TT
d
kA
TThA
dt
dT
mc
−+−−=
−−−−=
System equations:
mass of each wafer
convection to or
from the airstream
conduction across
the separating wall.
=m
101)0( TT =
202)0( TT =
Heat Transfer Example (3)
−= TTT 11*
Let and
−= TTT 22*
)(
2*1*1*
1*
TTbaT
dt
dT
−−−=
Then
Equation 1
Heat Transfer Example (4)
How do we solve an equation of the form
xA
dt
xd =
?Equation 3
Based on our prior experience with linear time-invariant ODEs we try
Heat Transfer Example (5)
where is the “identity matrix”:
00…10
00…01
I
for n-dimensional
Heat Transfer Example (6)
Recall from algebra that if the vector and matrix are known, the
solution to
zyC
=
z
C
y
Heat Transfer Example (7)
0)det( =− IrA
Hence one condition that Equation 6 must satisfy is
Here
−−−− brba
brba
Equation 7
Heat Transfer Example (8)
The solutions and are called eigenvalues. Real-valued
eigenvalues provide the rates of exponential decay or growth in the
motion, which we will henceforth call damping rate .
1
r
2
r
Heat Transfer Example (9)
Substituting in Equation 9 and using
ar −=
+−
+−
=)(
)(
bab
bba
A
Heat Transfer Example (10)
0
)1(
2
)1(
1=+−
Equations 10 and 11 are not independent. (They cannot be; their
determinant is zero.) They are essentially one equation
Equation 12
Heat Transfer Example (11)
Next we return to Equation 9, substitute the second eigenvalue
and solve for the eigenvector associated with it.
bar 2
2−−=
0
)2(
2
2
=
−−−
−−−
rbab
brba
Heat Transfer Example (12)
Again we have one equation and two unknowns. We set
and then, from Equation 15, and so the eigenvector
associated with the second eigenvalue, , is
1
)2(
1=
1
)2(
2−=
bar 2
2−−=
−
=1
1
)2(
Heat Transfer Example (13)
Setting in Equation 16 and applying Equation 17 we have
0=t
−
−
=
−
+
=
−
+
=
+−−
TT
TT
ccececx baa
20
10
21
)0)(2(
2
)0(
11
1
1
1
1
1
1
1
)0(
Heat Transfer Example (14)
Hence the solution to the ODEs in Equations 1 and 2 and the
initial conditions in Equation 17 is
Heat Transfer Example (15)
To what physical behavior do the eigenvalues and eigenvectors
correspond?
The first eigenvalue is the damping rate each wafer experiences as its
Homework Assignment 17
Read:
Chapter 8, Section 8.5.1.1
Work:
Short Quiz #2