Solve A(x) ∆F[f]/∆f + J(x)F[f] = 0

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Discussion Overview

The discussion revolves around the differential equation involving a functional F of f(x), specifically the equation A(x) ∆F[f]/∆f + J(x)F[f] = 0. Participants explore the nature of the equation, the dependencies of f, and the implications of treating F as a functional rather than a standard function.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether f must depend on x, indicating a potential ambiguity in the formulation.
  • Another participant suggests that if F were a function of x instead of a functional, a series solution could be applied, implying a different approach to solving the equation.
  • A participant clarifies that F is indeed a functional, which takes a function f(x) as input and outputs a number, emphasizing the complexity of solving the equation under this definition.

Areas of Agreement / Disagreement

Participants express differing views on the nature of F and its dependence on x, indicating that there is no consensus on how to approach the solution of the equation.

Contextual Notes

The discussion highlights the distinction between functionals and functions, which may affect the methods available for solving the equation. There are unresolved questions regarding the dependencies and definitions involved.

Karlisbad
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Differential equation??

Let be F a functional of f(x) and J(x) and A(x) a function, then can we solve this?:

[tex]A(x)\frac{\delta F[f]}{\delta f}+J(x)F[f]=0[/tex]

J and A are known functions and F[f] is an unknown functional satisfying the equation above.:confused: :confused:
 
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does f have to depend on x?
 
and are you sure you don't mean F(x)? becaue then all you need to do is substitute in a series solution.
 
F is a functional (a function of function :-p ) you introduce any function f(x) inside F and you get a number.. if F were a function i would know how to solve it...:redface:
 

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