Differential Equations
for Engineers: the Essentials
Supplement to Class 21 Notes
Systems of First Order ODEs
Coordinate Systems and
Cascaded Systems Example
Coupled and Uncoupled ODEs
“Coupled” ODEs (especially of higher order) are relatively
complicated to solve:
n
x
AAA
x
…
1
11211
1
x
A
x
0…0
1
11
1
while “uncoupled” ODEs are easy to solve
Coupled and Uncoupled ODEs (2)
The solution to the uncoupled ODEs is
Often (but not always) we can find a new coordinate system such that a
given coupled ODE is transformed into an uncoupled one. (We must
permit complex eigenvectors.)
Suppose we have a coupled system of first order ODEs of the form
)0(
0…0
)(
1
1
11
tA
x
e
tx
Matrix Diagonalization
Suppose the given nby nmatrix has nindependent eigenvectors .
Consider the matrix whose columns are the eigenvectors of .
where
A
T
A
)(i
r
0...0
1
Matrix Diagonalization (2)
Since, by hypothesis, the columns of the matrix are independent,
it is invertible, and so from
Then
and from
T
yTx =
xAx =
Matrix Diagonalization (3)
becomes
Recalling that
we see that we have found a coordinate system in which the
matrix is diagonalized and the variables are therefore uncoupled.
yDy =
r
0...0
1
A
Matrix Diagonalization (4)
Example: Cascaded first order systems
__
10
x
20
x
)(
1tx
)(
2tx
+
a
__
10
x
20
x
)(
1tx
)(
2tx
+
( )
0))((det 21 =++=−
rrIrA
Matrix Diagonalization (5)
a
Matrix Diagonalization (6)
( )
=
=−
0
0
)1(
)1(
2
)1(
a
IrA
( )
=+−
=−
a
IrA
0)(
0
)2(
)2(
)2(
2
)2(
112
)2(
Matrix Diagonalization (7)
−
=
/)(0
11
21
a
T
1
=+
CB
0
=+
ED
Matrix Diagonalization (8)
−
−−
=
−
)/(0
)/(1
21
21
1
a
a
T
The state vector for the diagonalized matrix is
It satisfies
111
yy
−=
Matrix Diagonalization (9)
Since most systems can be diagonalized, why don’t we do it as a matter of
course?
Because:
The math associated with diagonalization is about the same
difficulty as that needed for solving the problem in its
original form
Matrix Diagonalization (10)
−
−
=
2
1
0
0
2
1
0x
x
a
x
x
Consider now cascaded first order systems with the same eigenvalue:
021
==
Matrix Diagonalization (11)
The matrix is no longer diagonalizable. (Notice the terms
involving in in the previous example.) What
does the solution of the ODE look like? We can solve the
equations sequentially:
A
)/(121
−
1−
T
xx
202
−=
Matrix Diagonalization (12)
Hence the solution can be expressed as
This is of the form
)2(
2
)1(
1
)( xcxctx +=
as given in the discussion about repeated eigenvalues. Here
Matrix Diagonalization (13)
That we can diagonalize means that a system of nfirst order coupled
equations is equivalent to nuncoupled first order systems. We
have only to find the right coordinate system.
Among many other things, this tells us that we cannot usually
represent such a system with a single nth order equation of the
form
Matrix Diagonalization (14)
In summary:
We can diagonalize an nby nsystem matrix when the system has n
independent eigenvectors, which systems often do.
We typically do not solve ODEs by diagonalization because it does
not save work and it hides physical significance.