Linear vs Nonlinear: a debate
[1] Linearization near equilibrium
[2] Exponential Response Formula (first order)
[3] Potential blowup of solutions to a nonlinear equation
Hour Exam I Wednesday. Locations for both lectures:
10250 A  G
Walker H  Z
Office hours as usual on Wednesday
Review: Nonlinear vs Linear
y'(t) = F(t,y(t))
vs
r(t) x'(t) + p(t) x(t) = q(t)
This is in the form of a debate, between Linn E. R. (on the right)
and Chao S. (on the left). I'll take a vote at the end.
Linn: I'd like to begin by making the point that there is a solution
procedure for linear equations, which reduces solution of any linear
equation to integration: Multiply the equation through by a factor so that
the two terms become the two terms in
(d/dx)(ux) . Then integrate.
Sometimes you can just *see* this:
t^2 x' + 2t x = (d/dt)(t^2 x)
for example.
If we are in *reduced* standard form, so
r = 1 , then this can be done
systematically:
We seek
u(t)
such that
u (x' + px) =
(d/dt) (ux)
i.e.
pu = u' : separable, with solution
u = e^{\int p(t) dt}
(any constant of integration will do here). Then integrate both sides of
(d/dt) (ux) = uq
:
x = u^{t} int uq dt
The constant of integration is in this integral, so the general solution
has the form
x = x_p + c u^{1} .
Another lovely feature of linear equations is that the constant of integration
in the solution of a linear equation always appears right there.
The "associated homogeneous equation" is
x' + px = 0 : separable,
with solution
x_h = e^{\int p(t) dt} . Look! this is the reciprocal of
the integrating factor! Wonderful!
In most applications,
u{1}
falls off to zero
as
t
gets large; the term
c u^{1}
is a "transient."
Chao: That's a lot of integration .
...
I'm more interested in the general
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 Spring '09
 vogan
 Chaos Theory, Elementary algebra, Complex number, Chao S.

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