Re: Can't solve Diff Equ
- To: mathgroup at smc.vnet.net
- Subject: [mg42761] Re: [mg42732] Can't solve Diff Equ
- From: Selwyn Hollis <selwynh at earthlink.net>
- Date: Fri, 25 Jul 2003 05:08:06 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Is the function c[x] known? Regardless, you'll have better luck with
equ=D[x[t], {t, 2}] == D[c[x[t]], {x[t], 1}] D[x[t], {t, 1}]
DSolve[equ,x[t],t]
On the other hand, the substitution of x'=v and x'' = v dv/dx produces
the very simple equation
v'[x]==c'[x]v[x]
Hence, v[x_]:= C[1]*Exp[c[x]], which is the same as
x'[t] == x0*Exp[c[x[t]]-c[x0]]
By separating variables, you get
Integrate[Exp[c[x0]-c[z]], {z,0,x[t]}] == x0*t
So there are now two difficulties: computing the integral and, once
that's done, solving for x[t]. You'll notice that this is reflected in
the solution given by Mathematica.
-----
Selwyn Hollis
http://www.math.armstrong.edu/faculty/hollis
On Thursday, July 24, 2003, at 04:10 AM, 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
>
>