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18.03 Class 12, March 6, 2006
Homogeneous constant coefficient linear equations: complex or repeated
roots,
damping criteria.
[1] We are studying equations of the form
x" + b x' + k x
=
0
(*)
which model a mass, dashpot, spring system without external forcing
term.
We found that
(*)
has an exponential solution
e^{rt}
exactly when
r
is a root of the "characteristic polynomial"
p(s)
=
s^2 + bs + k
Example A.
x" + 5x' + 4x = 0 .
We did this:
The characteristic polynomial
s^2 + 5s + 4
factors as
(s + 1)(s + 4)
so the roots are
r = 1
and
r = 4 . The corresponding exponential
solutions
are
e^{t}
and
e^{4t} .
The general solution is a linear
combination
of these:
x = c1 e^{t} + c2 e^{4t} .
All solutions go to zero: no oscillation here. When the roots are real
and
not equal to each other the system is called "Overdamped."
Example B.
x" + 4x' + 5x = 0
The characteristic polynomial
s^2 + 4s + 5
has roots
r = 2 + sqrt(45) = 2 + i
Our old friend
i = sqrt(1) appears, and we have exponential solutions
e^{(2+i)t} ,
e^{(2i)t}
I guess we were expecting REAL valued solutions. For this we have:
Theorem:
If
x
is a complexvalued solution to
mx" + bx' + kx = 0,
where
m, b, and
k
are real, then the real and imaginary parts of
x
are also solutions.
Proof:
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 Winter '08
 Staff
 Linear Equations, Equations

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