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Re: differential equation with buondary conditions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg30323] Re: differential equation with buondary conditions
*From*: "Allan Hayes" <hay at haystack.demon.co.uk>
*Date*: Sat, 11 Aug 2001 03:39:46 -0400 (EDT)
*References*: <9kqk0f$4g6$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Gustavo,
We have,
DSolve[{y''[x]+k y[x]==0, y[0]==0,y[a]==0},y,x]
{{y -> Function[{x}, 0]}}
This is the generic solution, the only other possibility is for special
values of k and/or a (see below)
Leave out the boundary conditions
sol=DSolve[{y''[x]+k y[x]==0},y,x]
{{y -> Function[{x}, C[1]*Cos[Sqrt[k]*x] +
C[2]*Sin[Sqrt[k]*x]]}}
Get the function
ys = y/.sol[[1]]
Function[{x}, C[1]*Cos[Sqrt[k]*x] + C[2]*Sin[Sqrt[k]*x]]
The first boundary condition gives
ys[0]==0
C[1] == 0
Using this on the second boundary condition we get
ys[a]==0/.C[1]->0
C[2]*Sin[a*Sqrt[k]] == 0
This entails C[2] =0 unless Sin[a*Sqrt[k]] =0.
--
Allan
---------------------
Allan Hayes
Mathematica Training and Consulting
Leicester UK
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565
"Gustavo Seabra" <gseabra at swbell.net> wrote in message
news:9kqk0f$4g6$1 at smc.vnet.net...
> Hello,
>
> I'm trying to make Mathematica solve the following:
>
> y''[x] + k y[x] == 0
>
> subject to the boundary conditions:
> y[x<0] = 0
> y[x>a] = 0
> so that y[x] != 0 only if 0 < x < a.
> (yes, it's the "particle in a 1-d box problem.)
>
> If I just do: DSolve[{y''[x] + k y[x] == 0}, y[x], x]
> it works fine, giving:
> {{y[x] -> C[2] Cos[Sqrt[k] x] + C[1] Sin[Sqrt[k] x]}},
> which is perfectly ok.
>
> But if I include the boundary conditions y[0] == y[a] == 0,
> it doesn't work.
>
> Any ideas?
> --
> -----------------------------------------------------------------
> Gustavo Seabra - Graduate Student
> Chemistry Department
> Kansas State University
> -----------------------------------------------------------------
>
>
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