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Re: Solving for a function in an Integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg40824] Re: [mg40790] Solving for a function in an Integral
  • From: sean kim <shawn_s_kim at yahoo.com>
  • Date: Sat, 19 Apr 2003 23:03:36 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

hello. 

maybe the f[x] is actually f[x,t]?  when you try to take the derivative
of the f[x] in terms of t, mathemtica won't do it because f doesn't
depend on t. What seems like needs to be done is to change it to a pde?


In[1]:= D[ Integrate[Sin[x + m[t]], t] , t]
Out[1]= Sin[x + m[t]]

In[2]:= D[f[x], t]
Out[2]= 0

In[3]:= D[f[x, t], t]
this will give it a proper derivative of f[x,t] in terms of the t 

then solve the f'(x,t) == Sin(x + m(t)) for m(t).

it gives me errors 

In[6]:= Solve[{D[f[x, t], t] == D[ Integrate[Sin[x + m[t]], t] , t]},
m[t]]

so some solutions may not be found."

{{m[t] -> -x + ArcSin[f'[x,t]]}}

(I edited the eqn a bit since it doesn't copy and paste right.) 

or if the f[x]is indeed only dependent on x, and you are taking spatial
derivative of f[x] and then taking the time derivative of the rhs then,
it should look like this. 

In[7]:= D[f[x], x] == D[ Integrate[Sin[x + m[t]], t] , t]
Out[7]= f'[x] == Sin[x + m[t]]

which then can be solved again with same error about inverse function. 

hope that helps.

(my suggestions may not be as elegant or useful as others since i'm not
a mathematician, but let me know it helps) 


--- Steven Clarke <clarkesm at earthlink.net> wrote:
> Hi all
> 
> I'm wondering if there is a way to solve for a function within an
> integral.
> 
> Specificly, I have something like this:
> 
> d[x] == Integrate[Sin[x + m[t]],t]
> 
> so I have function d, dependent on x, which is equal to the integral
> of
> sin[x + m[t]] with respect to t, when m[t] is a function of t.
> 
> It looks sort of like a differential equation, but not really, and
> DSolve
> didn't like it.  I thought about taking the derivative of both sides
> with
> respect to t, but then the d[x] just goes completely away, and that
> can't
> work.
> 
> Is this problem mathmatically possible, and if so, what kind of
> problem is
> it (ie, ODE, Partial Differential Equations, something else) so I can
> go
> look up a book on those types of problems, and finally, can
> Mathmatica solve
> this type of problem?
> 
> Thanks
> 
> Steve
> 
> 


=====
when riding a dead horse,  some dismount.

while others... 
buy a new whip.

which one might you be?

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