Re: Can't solve Diff Equ
- To: mathgroup at smc.vnet.net
- Subject: [mg42777] Re: Can't solve Diff Equ
- From: Joel storch <jstorch at earthlink.net>
- Date: Fri, 25 Jul 2003 05:08:30 -0400 (EDT)
- References: <bfo4ca$cbk$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Oliver Friedrich wrote:
> Hallo,
>
> I try to solve the following differential equation:
>
> equ=D[x[t], {t, 2}] == D[c[x], {x, 1}] D[x[t], {t, 1}]
>
> DSolve[equ,x[t],t]
>
> Mathematica 4.2 returns it unsolved.
>
> 1. Does DSolve need any more information ?
>
> 2. What kind of DiffEqu is it, can't solve DSolve those kinds at all ?
>
> 3. Are there some special packages capable of "cracking" this equation ?
>
> To give some background information.
>
> An acoustic signal propagates through a medium with propagation velocity
> c[x], propagation depends on the location in the medium. I'm searching
> for a function x[t] which describes the location of the pulse as a
> function of time.
>
> Any ideas ? I guess, that this problem should be well known. Is this
> equation a well known one in this area?
>
> Oliver Friedrich
>
Make the substitution dx/dt=u and express D[x[t],{t,2}] in the form
u du/dx. You will then be able to integrate once and will obtain
u=c[x] + const.
It follows that
t=Integrate[1/(c[x]+const.),x]
Thats about as far as you can get in general. If you know the explicit
form of c[x], you may be able to perform the integration which
will lead to an implicit relation for x[t].