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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10
votes
1answer
121 views

What is the solution of the following functional equation? (I must confess it is a headache for me)

Find all the functions $f: \mathbb{Z} \to \mathbb{Q} $ such that $f(\frac{x+y}{3})=\frac{f(x)+f(y)}{2}$; $\forall x,y\in\mathbb{Z}$ knowing that $\frac{x+y}{3}\in\mathbb{Z}$.
3
votes
1answer
144 views

Functional equations leading to sine and cosine

This question is a possibly harder version of: Find $g'(x)$ at $x=0$. Question. Let $f,g :\mathbb R\to\mathbb R$, such that \begin{align} f(x-y)=f(x)\, g(y)-f(y)\, g(x), \tag{1}\\ g(x-y)=g(x)\, ...
0
votes
0answers
49 views

Non-constant solution of $f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$ in rings

The answers to this question prove that over an integral domain the functional equation $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ has only solutions where one of $f$ or $g$ is constant. The proofs make ...
3
votes
2answers
116 views

Find $g'(x)$ at $x=0$

The question is: Question. Let $$f(x-y)=f(x)\cdot g(y)-f(y)\cdot g(x)$$ and $$g(x-y)=g(x)\cdot g(y)+f(x)\cdot f(y)$$ for all $x,y \in \mathbb{R} $. If right hand derivative at $x=0$ exists for ...
2
votes
1answer
79 views

Functional Equation $f(n) = 2 f(n / f(n) )$

I have the following functional equation: $f(n) = 2 \cdot f\left(\frac{n}{f(n)}\right)$ Under the precondition that $ f(n) = \omega(1) $, monotinic and a initial value $ f(1) = \Theta(1) $ one can ...
5
votes
3answers
193 views

Continuous solutions of $f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$

Consider the following functional equation: $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ where the equation holds for all $x,y,z \in \mathbb{R}$. One solution is $f(x)=cx$ and $g(x)=1$. What are all the ...
0
votes
0answers
59 views

Probability distribution satisfying constraints?

Continuing from this question. Given two random variables $X$ and $Y$ where $X \sim \operatorname{Beta}(a, b)$ and $Y \sim \operatorname{Beta}(c, d)$, I'm looking for a random variable $Z$ with a ...
7
votes
4answers
190 views

Find all real to real function satisfy this functional equation.! $f((x+y)/(x-y))=[f(x)+f(y)]/[f(x)-f(y)]$

Find all real to real function satisfy this functional equation.! $$f\left(\frac {x+y}{x-y}\right)=\frac {f(x)+f(y)}{f(x)-f(y)}$$ I couldn't get to the final answer but I get $f(0) = 0$ and $f(1) = ...
7
votes
1answer
218 views

Find all continuous functions $ f:\mathbb{R} \rightarrow \mathbb{R}$ such that

Find all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x)^2 - 2\Bigl\lfloor\,2\,\bigl|f(2x)\bigr|\,\Bigr\rfloor f(x) + \Bigl\lfloor{4f(2x)^2-1}\Bigr\rfloor = 0\,.$$ Here ...
3
votes
2answers
297 views

Find a solution for f(1/x)+f(1+x)=x

Title says all. If f is an analytic function on the real line, and $f(1/x)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$? Additionally, what are any solutions for $f(1/x)-f(x+1)=x$?
8
votes
2answers
159 views

How find this function equation $(f(x))^2-(f(y))^2=f(x+y)\cdot f(x-y)$

Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that satisfying the function equation $$(f(x))^2-(f(y))^2=f(x+y)\cdot f(x-y)$$ By the way :I have see this problem( is ...
0
votes
0answers
36 views

Symmetric expressions for homogeneous functions

Suppose $f(x_1, \ldots, x_n)$ is a homogeneous function, i.e. a function such that $$f(\lambda x_1, \ldots, \lambda x_n) = \lambda^d f(x_1, \ldots, x_n)$$ for all $\lambda$ and for some positive ...
6
votes
3answers
267 views

How to solve this infinite set of equations?

If I can find a solution to the following set of equations then, with a bit of luck, I should be able to derive all sorts of nifty new results in non-equilibrium statistical mechanics. However, I'm ...
1
vote
1answer
55 views

Reference request: Where is this functional equation found?

$$ g\left(\frac{x+y}{1+xy}\right) = g(x)g(y). $$ One solution is $$ g(x) = \frac{1+x}{1-x}. $$ Another is $$ g(x) = \sqrt\frac{1+x}{1-x}. $$ Any other power of the first solution is also a solution, ...
7
votes
1answer
118 views

Periodic solution of differential equation y′=f(y)

Let $f∈C^∞(ℝ^2,ℝ^2)$ and $∀x∈ℝ^2$ $f(kx)=k^2f(x)$ for $k∈ℝ$ Show that any periodic solution of $y′=f(y)$ is constant. My attempt : Let $\lambda \in \mathbb{R}$. Let $g$ a periodic solution ...
2
votes
1answer
49 views

Existence of a function with a changing period

$f,\alpha$ are continuous $\mathbb{R}\to\mathbb{R}$ functions satisfying: $$f\big(x+\alpha(x)\big)=f(x)$$ If $f$ is non-constant, must $\alpha$ be constant? My idea was to use the fact ...
0
votes
0answers
24 views

Question on 2 functional equations.

Let $z,x$ be complex numbers. Im looking for analytic functions $f(z)$ such that : $$1) \exp(\ln^{5} (f(x))=\sum_i a_i f(b_ix)$$ $$2)f(x)^5=\sum_j c_j f(d_jx)$$ holds for all $x$ and where both ...
2
votes
2answers
40 views

Find all functions

Hi I'm not sure if I'm correct in this example: Find all functions such that $f(x-|x|)+f(x+|x|)=x$ where $x \in R$, so my answer is the are only one function satisfying this condition ...
3
votes
1answer
41 views

Functional equation with function defined on $\mathbb{N}^{*}$

Let $ f:\mathbb{N}^{*} \mapsto \mathbb{N}^{*} $ be a function with the following property: $$ \frac{f(x+1)f(x)-2x}{f(x)}=\frac{2f^2(x)}{x+f(x)}-1$$ Determine all functions with this property. (I'm ...
4
votes
5answers
153 views

Is there an everywhere-defined function that satisfies $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$

Is there a function $f:\mathbb{R}\to\mathbb{R}$ which is differentiable and satisfies the following: (1) $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$ (2) $f'(0)=1$ (1) is the functional equation for ...
0
votes
0answers
46 views

What is the analytical solution to a Volterra integral equation?

I need to solve a following equation: \begin{equation} r_{k+1} = -\sum\limits_{l=0}^{k-1} r_l \cdot (k-l) \cdot \left(\frac{\omega}{t_c - l}\right)^{2 \beta} + \delta_{k,0} \end{equation} subject to ...
1
vote
1answer
163 views

My proof of Cauchy functional equation?

Although I have not quite studied functional equations, I came upon Cauchy functional equation and tried to prove it. Here is what I have done: We are given the condition, $f(x+y)=f(x)+f(y)$. ...
3
votes
1answer
157 views

Real analytic $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the real analytic functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
7
votes
2answers
248 views

Solving the differential equation $f'(x)=af(x+b)$

How does one find all the differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ which satisfy the equation $$ f'(x)=af(x+b),\quad \text{for}\quad a,b \in \mathbb{R}? $$ I see that functions ...
8
votes
2answers
182 views

Continuous $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the continuous functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
0
votes
0answers
65 views

Numerically solving delay differential equations

I understand that the Dickman Function can be solved numerically by converting it to a delay differential equation. The Mathworld link above describes how this may be achieved for $$\rm ...
3
votes
1answer
57 views

Weird ordinary differential equation

I am searching for the functions $f,g:[0,\infty) \to [0,\infty)$ which are both increasing, $f'$ is strictly increasing, $g$ is the inverse of $f$ and $$ g(x)^2=f(x)^2\cdot g'(x)$$ My approach was ...
1
vote
1answer
29 views

Functional equation question

The following has come up in the course of my research. I'm looking for a function $f:\mathbb{Z^\star}\to\mathbb{R}$ such that $$ 2f(i) - f(i+j) - f(i-j) = \lambda j $$ for all $i\ge0$ and all $j$ ...
0
votes
1answer
45 views

Solution $\chi$ of $\chi_k(x)=\chi_x(f_k(x))$ given $f_k$ bijective

Given a family $\{ f_a \}_{a\in H}$ of bijective functions over a set $H$ I need to find a family of functions $\{ \chi_a \}_{a\in H}$ from $H$ to $H$ such that for every fixed $k$ in the family ...
4
votes
3answers
90 views

Retrieving the Taylor series for $\log$ from its functional equation

Consider the unique continuous function $\mathbb{R}^+\to\mathbb{R}$ such that: $$f(xy)=f(x)+f(y),\qquad f(e)=1$$ where $\displaystyle e=1+\sum_{n=1}^{\infty} \frac{1}{n!}$. Assuming $f$ has a ...
5
votes
5answers
130 views

Is there analytic solution to $x^y=y^x\land x\neq y$ as $y(x)$?

Equation $x^y=y^x\land x\neq y$ has trivial solution $ y(x) = x$. Is there non trivial solution given say in terms of elementary or special functions as $y(x)$? A solution that would yield $y(2) = 4$ ...
0
votes
1answer
27 views

Finding polynomials sattisfying $P\bigr(-c + K/(u+c)\bigl) (u+c)^2/K =P(u)$

Is there any simple way to find the polynomials satisfying the functional relation \begin{align*} P\left(-c + \frac{K}{u+c}\right) \frac{(u+c)^2}{K} = P(u) \tag{*} \end{align*} Where $K = ...
4
votes
2answers
126 views

How to solve $(f'(x+1)+f'(x-1))f(x)-(f(x+1)+f(x-1))f'(x)=0$

$$(f'(x+1)+f'(x-1))f(x)-(f(x+1)+f(x-1))f'(x)=0$$ I don't have any ideas about the solution of this problem. How can I solve this differential equation?
17
votes
1answer
342 views

Find $f(x)$ where $ f(x)+f\left(\frac{1-x}x\right)=x$

What function satisfies $ f(x)+f\left(\frac{1-x}x\right)=x$ ?
1
vote
2answers
108 views

show that $f(x)=c\log x $ for some $c$

Let, $f$: $\mathbb{R^+}$$\rightarrow$$\mathbb{R}$ be a continuous function satisfying $f(xy)=f(x)+f(y)$. Prove that, $f(x)=c\log x$ for some $c>0$.
1
vote
0answers
108 views

Galois Theory for Differential Equations?

Consider the set of algebraically primitive functions: that is the set of functions who can be created through some recursive expression involving arithmetic operators. Example $$y = x +1$$ $$y = ...
3
votes
5answers
90 views

Simple functional equation

I have a simple functional equation: $$ \alpha(x) = {1 \over 2}\left[\alpha\left(x - 1\right) + \alpha\left(x + 1\right)\right]\,, \qquad \alpha\left(0\right) = 1\,,\quad\alpha\left(m\right) = 0 $$ I ...
2
votes
1answer
94 views

How find this function $f(2^m)=?$

show that:there exists unique function $f:N^{+}\to N^{+}$,such $$f(1)=f(2)=1$$ and $$f(n)=f(f(n-1))+f(n-f(n-1))),n=3,4,\cdots$$ and Find $f(2^m),m>2,m\in N$ My try: let $n=3$ ,then we have ...
7
votes
1answer
179 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
19
votes
2answers
603 views

$\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

I need to find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that ...
2
votes
1answer
129 views

Characterization of arithmetic mean

Let $f_m$: $\mathbb{R}_{\geq 0}^m \to \mathbb{R}_{\geq 0}$ be a series of functions that satisfy symmetry (when permuting indices), strong monotonicity (in every entry), homogeneity of degree 1, ...
1
vote
2answers
70 views

functional equations with restricted domain

How can I find all discontious solutions of functional equation $f(xy)=f(x)f(y)$ on $[0,1]$. Similar question is to find all solutions of the equation $f(x+y)=f(x)+f(y)$ on $[0,\infty)$. Can we still ...
1
vote
6answers
208 views

Solutions of $f(x)\cdot f(y)=f(x\cdot y)$ [duplicate]

Can anyone give me a classification of the real functions of one variable such that $f(x)f(y)=f(xy)$? I have searched the web, but I haven't found any text that discusses my question. Answers and/or ...
6
votes
1answer
243 views

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfied that: $f(xy)+f(x+y)=f(xy+x)+f(y)\quad\forall x,y \in \mathbb{R}$

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfied that: $$f(xy)+f(x+y)=f(xy+x)+f(y)$$ $\forall x,y \in \mathbb{R}$ I have tried that : $P(y;x)-P(x;y)$: ...
9
votes
3answers
147 views

Solutions for the Functional Equation $f(x^2)=f(x)^2$

Suppose a continuous function $f:[0,1]\rightarrow [{0,1}]$ satisfies the functional equation $f(x^2)=f(x)^2$. Then I conjecture that we must have $f(x)=0$ or $f(x)=x^r$ for some real number $r\geq 0$. ...
2
votes
2answers
39 views

function equation with translation of independent variable

The following has come up in some work I'm doing: If $\frac{f(x+a)}{f(a)}=g(x)$ , where $g(x)$ is given and $a \geq 0$ is a constant, what is $f(x)$ ? We can assume that $g(x)>0 ~ \forall x$ . Of ...
5
votes
3answers
73 views

Simple Functional Equation $\frac{f(a)-f(b)}{a-b}\cdot(-a)+f(a)=-ab$

Compute all real-valued functions $f$ so that the line between any two points on the graph $f$ intersects the $x$-axis at the product of those two points' $x$-coordinates times $-1$. (if we do some ...
1
vote
1answer
53 views

Solving limit equations

Notice the following: all first order differential equations take on the form: $$G(x,f(x),\frac{d}{dx}[f(x)]) = 0 $$ notice that we can replace $\frac{d}{dx}[f(x)]$ with the expression $$ ...
8
votes
2answers
167 views

What do logarithms distribute over?

I notice that division distributes over addition Root extraction distributes over multiplication What operator do logarithms distribute over: ie: what non-constant function $H \in C^2 \rightarrow C$ ...
4
votes
2answers
71 views

finding the value of $f(\frac{1}{7})$

$f$ is a function mapping positive reals between $0$ and $1$ to reals. Let $f$ be given by, $f( \frac{x+y}{2} ) = (1-a)f(x)+af(y)$ where $y > x$ and $a$ being a constant. Also,$f(0) = 0$ and $f(1) ...