# Tagged Questions

The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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### Two variable equation, weighting of stock

I am trying to find the weighting of stocks in a portfolio (both variables should be between 0 and 1 but added together should be 1). the equation is: $1.2353 = x(1.2) + y(0.9)$ I have it simplified ...
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### How can I find $f'(2x)?$

it might seem a little bit elementary. $f$ is defined on $\Bbb R$ and it is differantiable. and is not equal to zero. if $xf(x)-yf(y)=(x-y)f(x+y)$ then find what is $f'(2x)$ equal to?. ...
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### For which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$?

I am wondering for which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$? I am quite sure a complete characterisation will be very hard, but I'm looking for partial ...
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### What “natural” functions besides $g(x) = x$ and $g(x) =\dfrac{c^{x}}{c^{x}+c^{1/2}}$ satisfy $g(x)+g(1-x) = 1$?

I'm not quite sure how I should state this question. This is one way: What "natural" functions besides $g(x) = x$ and $g(x) =\dfrac{c^{x}}{c^{x}+c^{1/2}}$ satisfy $g(x)+g(1-x) = 1$? By "natural" I ...
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### A function that is zero for every integer multiple of $k$

I'm new to this kind of question so it may be a trivial one but I can't find a general solution: I'm searching for a function that has a value of $0$ for every integer multiple of some fixed real ...
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### Find the function $f(x)$

Given $f(x)$ is a differentiable function such that $$f(x+y)=e^xf(y)+e^yf(x)$$ $\forall$ $x,y$ $\in$ $\mathbb{R}$ and $f'(0)=1$. Find $f(x)$ if we put $y=0$ we get $$f(x)=e^xf(0)+f(x)$$ $\implies$ ...
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### Find continuous functions that satisfy $f(f(x))=x$ over the reals.

I'm looking for a method to solve: $$f(f(x))=x$$ Where $f$ is defined for $x \in R$ So far by inverting both sides I have: $f(x)=f^{-1}(x)$ Which means that my function should be symmetrical over ...
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### How to solve this nonlinear functional recurrence

I study two similar nonlinear functional recurrence systems, given by $$P_\pm:\qquad f_n\cdot(1\pm g f_{n-1}) = g\mp(1+2g)f_{n-1} \qquad (n>0)$$ and $$f_0=g$$ Here $f_n$ and $g$ are functions of ...
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### Find a function $f$ which satisfies $f(mn) = f(m)f(n)$ for positive integers $m,n$ and $f(2)=2$

We are to find a function f which follows the following properties $$f(mn)=f(m)f(n),\; f(2)=2.$$ I can easily find all the values of $f(2^n)$ but I am confused on how to find for the odd numbers and ...
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### Why does the Lagrange equation have to be zero?

I know it's a pretty basic question, but I still don't get it since starting Lagranian mechanics this year. I tried to read Stone and Goldbart's "Mathematics for Physics" and they said: Suppose ...
### $f(x) \ge f(x + \sin x)$, nonconstant functions, infinite number of solutions to $f'(x) = 0$.
Let $\mathcal{F}$ be the set of all the differentiable functions $f: \mathbb{R} \to \mathbb{R}$, which have the property $f(x) \ge f(x + \sin x)$, for all $x \in \mathbb{R}$. Prove that ...