Re: form equation takes at large value of time

In article <1117455120.690141.262460@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
paul <pjoseph@xxxxxxxxx> wrote:


>I have a question re. whether anything can be said about this
>equation's behaviour at large values of time, purely by inspection.


>The differential eqation is this:

>y'' +Cy' + A*exp(-Bt)y = 0

>where A, B, C are constants, y is say displacement and t is time.

>My question is this:

>At time t = infinity, is it correct to say that the system represented
>by this equation can be described by:

>y'' + Cy' = 0

>Based on the physical fact that it means that the "spring" has one to
>zero, I suspect the answer is yes, but would like to confirm.

Yes, but I don't think I'd call it "by inspection". I'm assuming your
A,B,C are all positive.

In fact, a fundamental set of solutions is
exp(-C t/2) J_{C/B}(2 sqrt(A)/B exp(-B t/2))
and
exp(-C t/2) Y_{C/B}(2 sqrt(A)/B exp(-B t/2))

where J_{C/B} and Y_{C/B} are the Bessel functions of the first
and second kinds of order C/B. The behaviour as t -> infinity
can be obtained from the behaviour of J_{C/B}(x) and Y_{C/B}(x) at
x=0: J_v(x) ~ const x^v and Y_v(x) ~ const x^{-v}. So our first
solution is asymptotic to a constant multiple of exp(-C t) as
t -> infinity and our second is asymptotic to a constant. And
this, of course, is also what happens with y'' + C y' = 0.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada



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