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Re: Solving for a function in an Integral

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  • Subject: [mg40826] Re: [mg40797] Solving for a function in an Integral
  • From: sean kim <shawn_s_kim at yahoo.com>
  • Date: Mon, 21 Apr 2003 06:52:12 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

i didn't realize there was another and later post on this. 

> f[x] = Integrate[Sin[x + m[t]],t] 
> That is, I have a function f, in x, which is equal to a integral of a
> function in x and t, integrated over t.

If the f[x] is equal to integral of a function in [x,t] then the
integral has to be both x and t dependent.  (intergrating a function in
terms of t when the fucntions has x and t doesn't get rid of the x
variable. )

this makes the original f[x] not f[x], but f[x,t].

> 1) Can this even be solved?
it appears that way. but it seems to be nonlinear... 

> 2) Does this kind of problem have a name (ie ODE, PDE, something
> else) so I can go look up a book on it something?

one you take the derivative of f[x,t] in t, then do the same withthe
right hand side, then it would seem you have an oridnary differential
equation as follows,

df(x,t)/dt = Sin(x+m(t))

i'm new to analysis of the differential equations, so my understanding
is a bit thin. but as i far as i know, the differential equations are
categorized by the order, number of differentials in terms of
independent variable, linearity. 

order just refers to the highest order of driative in your equation. (
yours is first)

number of differentials determine whether a given de is a partial or
ordinary de. ( your is ode)

nonlinearity of de is still a puzzle for me. Yours looks nonlinear, but
i can't tell you exactly why. 

i recommend any books on differential equations analysis using
Mathematica. there are so many of them out there. 

> 3) Can it be solved in Mathmatica, and if so, how?

this takes the derivatives ( both f[x,t] and integral in repsect to t. 

D[f[x, t], t] == D[Integrate[Sin[x + m[t]], t], t]

this takes derivative of f[x,t] in x and then integral in t.

D[f[x, t], x] == D[Integrate[Sin[x + m[t]], t], t]

i'm not sure which you need. 

they can be solved following ways. 
first with the derivatives in t 
Solve[ {D[f[x, t], t] == D[Integrate[Sin[x + m[t]], t], t]}, m[t]]

or derivative of f[x,t] in x. 

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


both solutions look about the same and like i posted in the other
message has some errors. 

it looks like an intersting problem.

hope things work out. 

sean from UCIrvine



=====
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while others... 
buy a new whip.

which one might you be?

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