Re: Solving for a function in an Integral
- To: mathgroup at smc.vnet.net
- 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 ===== when riding a dead horse, some dismount. while others... buy a new whip. which one might you be? __________________________________________________ Do you Yahoo!? The New Yahoo! Search - Faster. Easier. Bingo http://search.yahoo.com