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
52 views

Solutions to $g(ab) = ag(b) + bg(a)$ - “Zero function question”

This question was recently asked and then voluntarily removed by its author. I thought it was interesting enough to get an answer. The question was as follows: "The function $g:\mathbb R\to\mathbb ...
3
votes
3answers
89 views

Functional Equation : $f(x) = f(x + y^2 + f(y))$

This problem is from my textbook: Given : $f:\mathbb R\to\mathbb R$ Solve this functional equation : $f(x) = f(x + y^2 + f(y))$ I think this function is just a simple constant, so I try all my ...
1
vote
1answer
18 views

Finding Inverse of Function With Two Instances of X

I need to find $f^{-1}(2)$ where $f(x) = 2 + x^2 + tan(πx/2)$ I know can substitute $f(x)$ with $y$ and swap $x$ and $y$: $$x = 2 + y^2 + tan(πy/2)$$ But I'm having trouble eliminating the tangent: ...
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0answers
54 views

Fourier transform of a function involving $\sec(\omega)$

The summary of my question is: What should I make of $\mathcal{F}^{-1}\left[\frac{\csc(\omega)}{\omega^2-\beta^2}\right]$ where $\mathcal{F}^{-1}$ is the inverse fourier transform (taking $F(\omega)$ ...
4
votes
2answers
50 views

How to take partial derivatives of functions whose inputs depend on the same variable?

I am starting to learn about the Calculus of Variations and the Euler-Lagrange equation is extremely confusing to me: The Euler–Lagrange equation, then, is given by ...
1
vote
1answer
74 views

A fairly difficult differential equation

I was thinking what if I had a differential equation of the form: $$\frac{d^2u}{dx^2} + vu(x) = 0 $$ where $v(y(x))$, that is $y$ is a function of $x$. What are the possible restrictions that I can ...
12
votes
4answers
466 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 ...
8
votes
2answers
263 views

How to find all polynomials P(x) such that $P(x^2-2)=P(x)^2 -2$?

I am trying the fallowing exercise : Solve $P(X^2 -2)=P(X)^2 -2$ with P a monic polynomial (non-constant) My attempt : Let P satisfying $P(X^2-2) = (P(X))^2-2$ Then $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
27
votes
2answers
536 views

How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

Prove that there is no continuous functions $f:[0,1]\to R$ such that $$ f(x)+f(x^2)=x,x\in [0,1]. $$ My try: Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ ...
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vote
1answer
49 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
0
votes
0answers
22 views

Linear-like function

Suppose we need to find all the functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that $\forall x,y,z\in \mathbb{R} ~~ f(x+z) - f(x) = f(y+z) - f(y)$ It can be shown that $~~\forall r \in ...
0
votes
0answers
19 views

Mellin transform and a proof of the functional equaton for the $ \zeta (s) $

i would like to obtain a proof of the functional equation for the RIemann zeta function to do so i would like to know if there is a function or a distribution so $$ \int_{0}^{\infty}f(t)t^{s-1}dt = ...
0
votes
1answer
49 views

Worded Problem: Model a Plane Landing

I have no idea to model this. All I know are the two points $(50, 10)$ and $(0,0)$ Then from after solving I get $a=1/12500$ and $b=0$ The textbook answers are:
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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
135 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
48 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
78 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
186 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
51 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
170 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
214 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
283 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
155 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
33 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
265 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 ...
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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
108 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
48 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
152 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
42 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
141 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
153 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
233 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
63 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
56 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
28 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
89 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
128 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
338 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
103 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 $c>0$.
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vote
0answers
104 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
86 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 ...