Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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Finding twice-differentiable $f$ such that $f(x) = x^{-a} - 1 + f(1) + f'(1)(x-1)$

$a$ is a constant, $x \in (0,\infty)$. I encountered this differential equation playing around with some material, unsure if it hides something interesting. In fact, plugging $f(1)$ returns an ...
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Functional derivative of a function with nested integrals

I'm trying to solve a calculus of variations problem to find the cross-sectional area of a bar as a function of its length, which minimises its volume but has some fixed displacement at the free end. ...
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Can a function $f$ exist such that $f(x)f(y) = xy + 1$ for positive real $x$ and $y$

Problem. Can a function $f$ exist such that $f(x)f(y) = xy + 1$ for positive real $x$ and $y$ If not, is there something like a next best thing, perhaps only for integers, finite field, or maybe only ...
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calculate the terms of a function using base 2, number theory

I have the following problem from the book Teoria dos Numeros:um passeio com primos e outros numeros familiares pelo mundo inteiro. Let $f : \mathbb{N} >0 → \mathbb{N}$ be a defined function of the ...
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Extremal of a functional with one variable endpoint

Consider the functional $$J[y] = \int \limits_{0}^{b} {\frac{\sqrt{y'^2 + 1}}{y} \text{d}x}$$ with $y(0) = 0$ and the other endpoint somewhere along the circle $(x-9)^2 + y^2 - 9 = 0$ (call this ...
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A neat functional equation $f(x)=f(x+c)-f'(x+c)$

While investigating I thought of this functional equation $$f(x)=f(x+c)-f'(x+c)$$ and I wondered if there are any real analytic non-constant solutions. Here $c$ is some constant. Are there any real ...
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