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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1answer
47 views

$f(x)=xf(1)$ Doubt

I just started learning Functional Equations and I was working on a problem that asked me to find all functions satisfying a certain condition, and at some point I got $f(x)=xf(1)$, is there a way to ...
0
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1answer
123 views

If $f(2x)=2xf'(x)$, then find $f(x)$

If $f(x)$ is Analytic functions on $R$,and such $$2xf'(x)=f(2x)$$ Find all $f(x)$ My idea: let $$f(x)=\sum_{n=0}^{\infty}a_{n}x^n$$ so I can't Thank you
3
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3answers
153 views

$1992$ IMO Functional Equation problem

The problem states: Let $\Bbb R$ denote the set of all real numbers. Find all functions $f : \Bbb R \rightarrow \Bbb R$ such that $$f(x^{2}+f(y))=y+(f(x))^{2} \space \space \space \forall x, y \in ...
5
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2answers
87 views

Follow on from previous question: Functional Equation - a little tricky

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$. The answer to this has already been posted, but it doesn't explain why ...
83
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6answers
3k views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?

I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a continuous ...
7
votes
4answers
155 views

$f:\Bbb Z\to\Bbb Z$ such that $f(f(n)-2n)=2f(n)+n$ for all $n\in\Bbb Z$

Does there exists a function $f:\Bbb Z\to\Bbb Z$ such that $$f(f(n)-2n)=2f(n)+n$$ for all $n\in\Bbb Z$
2
votes
2answers
188 views

Recurrence relations on a continuous domain

While attempting to read Shannon's paper I came across the following (p. 3): suppose $N\colon \mathbb{R} \to \mathbb{R}$ is a function, which for some fixed (given) set of values $t_1, t_2, \dots, ...
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0answers
29 views

General solution of the recurrence equation with real shifts

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
7
votes
0answers
60 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really ...
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1answer
43 views
0
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0answers
22 views

Does there exists a function f:Z→Z such that f(f(n)−2n)=2f(n)+n for all n∈Z [duplicate]

Does there exists a function $f:Z→Z$ such that $f(f(n)−2n)=2f(n)+n$ for all $n∈Z$
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1answer
31 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
0
votes
1answer
109 views

More general reflection formula for Gamma function

It is know that $\Gamma(z)\, \Gamma{(1-z)}=\pi \csc( \pi z)$. Is there any formula for $\Gamma{(a+z)}\Gamma{(a-z)}$ where $a$ is a rational number, i. e., $a=p/k$ with $p, k$ integers and $z$ is ...
4
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1answer
93 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
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0answers
23 views

Multiplicative functional equation on Gaussian integers

Using polar form of complex numbers, it can be checked that all solutions $f:\Bbb C \to \Bbb R$ of the functional equation $$ f(zw)=f(z)f(w) \tag 1 $$ are of the form $$ ...
4
votes
1answer
60 views

Trivial funcional equation

I'm sorry and a little ashamed to ask this very simple question. The problem is that I'm not very familiar with functional equations (I just know that they can be tricky). The question is: which are ...
1
vote
1answer
27 views

Linear after change of variables

Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, ...
3
votes
1answer
60 views

Solving functional equation 2

Problem: find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2+f(y))=(f(x))^2+y^4 +2f(xy),\ \ \ \forall x,y\in\mathbb{R}$$
4
votes
1answer
56 views

Solving functional equation 1

find all functiions $f:\mathbb{R}\to\mathbb{R}$ such that $f'$ exists and $$f(x)=f\left(\frac{x}{2}\right)+\frac{x}{2}.f'(x),\forall x\in\mathbb{R}$$
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votes
1answer
41 views

Solve a functional equation

Find all functions $f:[0,+\infty)\to [0,+\infty)$ such that $f(x)\geq \frac{3x}{4}$ and $$f(4f(x)-3x)=x,\forall x\in[0,+\infty)$$
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0answers
9 views

Quicker way of solving Fout = Fref * A / B / C for a given Fout and Fref

I've got a chip device that calculates frequency based on the following equation: $Fout = Fref * A / B / C$ Where: $Fout$ is the output frequency $Fref$ is a reference frequency 0 < $A$ <= ...
2
votes
1answer
57 views

Solution to a Functional Equation $g'(x)=\dfrac{g(bx)-g(x)}{(b-1)x}$

What approaches can we take to solve the functional equation, $g:\mathbb{R}\to \mathbb{R}$ is a differentiable function, such that, $$g'(x)=\dfrac{g(bx)-g(x)}{(b-1)x}$$ Where, $b \in \mathbb{R}$ ...
16
votes
2answers
520 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 ...
6
votes
0answers
79 views

How to prove subadditive function?

Let $f: [0, \infty) \to \Bbb R$ be a function satisfying the following conditions: (1) For any $x,y \geq 0, f(x+y) \geq f(x) + f(y)$. (2) For any $x \in [0,2], f(x) \geq x^2 - x$. Prove that, for ...
12
votes
4answers
465 views

Find all bijections $\,\,f:[0,1]\rightarrow[0,1]\,$ satisfying $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the functional relation $$ f\big(2x-f(x)\big)=x, \tag{1} $$ for all ...
2
votes
1answer
33 views

continuous functions on rationals

Let $\Bbb Q$ be the set of rationals and $f:\Bbb Q\to \Bbb R$ be a continuous function. Then $f$ is bounded on some interval? If not, what happen if in addition $f$ satisfies $f(xy)=f(x)f(y)$ for all ...
7
votes
1answer
103 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 ...
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0answers
28 views

I tried this problem in the following way; is it right?

Let, f(x) be a twice differentiable function defined on (-1, 1) and f(0) = 1. Let, f(x) ≥ 0, f'(x) ≤ 0 and f''(x) ≤ f(x) for all x ≥ 0. Show that, f'(0) ≥ -√2. I am telling you what I did. First, ...
1
vote
0answers
52 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
6
votes
2answers
106 views

Show that $f(x) = x$ if $f(f(f(x))) = x$.

If $f: \mathbb{R} \to \mathbb{R}$ and $f$ is strictly increasing, show that $f(x) = x$ if $f(f(f(x))) = x$. So this compulsorily ESTABLISHES that $f(x) = x$ only, and no other solution. So, merely ...
2
votes
2answers
63 views

How do you find two functions $f$ and $g$ such that $f(x) \cdot g(x)=f(x)-g(x)$?

This was inspired by this question ( Logarithms with trigonometric inequality ) I already know the answer ( $f(x)=\tan^2 x$ and $g(x)=\sin^2 x$). However I am interested in how to find this answer. ...
10
votes
1answer
126 views

An IMO inspired problem

This problem from IMO 1988 is said to be one of the most elegant ones in functional equations. Problem : The function $f$ is defined on the set of all positive integers as follows: \begin{align} ...
2
votes
1answer
32 views

About the space $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$

I am reading Taylor's Partial differential equations III (nonlinear equations) (Section 1 of Chapter 16, Theorem 1.2), and Sogge's Lectures on Non-linear wave equatuions. I notice that in the energy ...
0
votes
0answers
17 views

Logarithmic functional equation on positive rational numbers

Let $\Bbb Q^+$ be the set of positive rationals and $f:\Bbb Q^+ \to \Bbb R$ be a continuous function satisfying $$ f(rs)=f(r)+f(s) $$ for all $r, s\in \Bbb Q^+$. Then $f$ has only the form $$ f(r)=c ...
0
votes
1answer
37 views

Multiplicative function on rationals [duplicate]

Let $\Bbb Q^+$ be the set of positive rational numbers. Find all solutions $f:\Bbb Q^+ \to \Bbb R$ of the functional equation $$ f(xy)=f(x)f(y), \quad x, y\in \Bbb Q. $$ Is $f(x)=x^a$ the only ...
0
votes
0answers
24 views

Solving differential functional equations with a restricted solution

I have a variable vector $X=\{x_1,x_2,...,x_n\}$, and a constant vector $V=\{v_1,v_2,...,v_n\}$. $f(x_i,X)$ is a function that takes X and xi as the parameter, for example: $f(x_i,X) = ...
1
vote
1answer
32 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants ...
1
vote
2answers
50 views

A scaling functional equation

I want to find a closed form of a function satisfying $$G(4z)=\frac{G(z)}{2z},$$ unfortunately I am not experienced with scaling problems, so I have no idea and would be thankful for any hints. ...
4
votes
2answers
54 views

Solve functional equation [closed]

find all functions $f:\mathbb{R^{*}}\to \mathbb{R}$ such that $$f(y^2f(x)+x^2+y)=x(f(y))^2+f(y)+x^2,\;\forall x,y\in \mathbb{R^{*}}$$ ($\mathbb{R^{*}}=\{x\in\mathbb{R},x\ne 0\})$
2
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0answers
51 views

Solving functional equation

Problem:find all continuous functions $f:[0,+\infty)\to [0,+\infty)$ such that $$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\;\forall x,y\in [0,+\infty)$$
1
vote
1answer
40 views

Functional equation with $f(1)$ integer

Here is a nice problem: Let $f:R\rightarrow R$ be a function, R is the set of real numbers, satisfying the following properties: $ f(1)$ is an integer and $xf(y)+yf(x)=(f(x+y))^2-f(x^2)-f(y^2)$, for ...
4
votes
2answers
95 views

Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x).

Problem : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x). My approach : The given equation can be written as $$(x-y)f(x+y) -(x+y)f(x-y) =4xy(x-y)(x+y)$$ $$\Rightarrow ...
-1
votes
1answer
69 views

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$? [duplicate]

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$?
5
votes
5answers
138 views

$\forall x\in\mathbb R$, $|x|\neq 1$ it is known that $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$. Find $f(x)$.

$\forall x\in\mathbb R$, $|x|\neq 1$ $$f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$$Find $f(x)$. Now what I'm actually looking for is an explanation of a solution to this ...
2
votes
2answers
159 views

Functional equation: $f\left(\frac{x-1}{x}\right)+ f\left(\frac{1}{1-x}\right)= 2- 2x$

There is a function given $f\left(\dfrac{x-1}{x}\right)+ f\left(\dfrac{1}{1-x}\right)= 2- 2x ,f\colon \Bbb R\setminus\{0,1\}\to \Bbb R$ How many fuction exist? I have no idea how to start
1
vote
0answers
42 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
1
vote
2answers
107 views

On a function with a (complicated) functional equation.

Let $g(x,y)$ be a function such that: I. $-1\lt g(x,y)\lt1.$ II. $$\ln(\frac{1+g(x,y)}{1-g(x,y)})+2y\tan^{-1}(yg(x,y))=2(y^2+1)x,$$ for $x\in\mathbb R, y\gt1.$ Then i. Show that $g(x,y)$ ...
7
votes
2answers
229 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 ...
3
votes
1answer
134 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

Does there exist a nontrivial cumulative distribution function $F$ on $\mathbb{R}_+$ for which $(F^2)'= (F')^{\ast k}$ for some $k > 0$?

This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the ...