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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7
votes
2answers
75 views

Solve the functional equation $ \dfrac{f(x)}{f(y)}=f\left( \dfrac{x-y}{f(y)} \right) $

Solve the functional equation $$ \frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right), $$ here $f: \mathbb{R} \to \mathbb{R}$ and $f$ is differentiable at $x=0.$ By set $x=y$ we get $f(0)=1$. ...
7
votes
3answers
134 views

Functions proof.

Find all functions $$f: \mathbb{Z} \rightarrow \mathbb{Z}$$ such that $$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$ for all integers $$a, b, c$$ satisfying $$a+b+c=0$$ I have no idea how to ...
8
votes
4answers
109 views

Is $f(x) = Cx\log x$ the only solution to $f(xy) = xf(y) + yf(x)$?

I was studying $L(x) = x \log x$ function and found that it satisfies the following functional equation for positive $x, y$: $$ f: \mathbb R^+ \to \mathbb R\\ f(xy) = x f(y) + y f(x) $$ I have a ...
3
votes
0answers
59 views

Solving an equation for a function

I am trying to do some proof and in connection with that this question arose: Can you find a decreasing function so that $$ 1-\frac{f(x)}{f(ax)} = (1-a)^2x^2 $$ where $0\leq a \leq 1$ and $x$ is ...
1
vote
0answers
33 views

Solve the system of functional equations.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \begin{cases} f(x(1+f(x)))=f(x)^2,\\ f(x(1-f(x)))=f(x) f(-x),\\ f(x(-1+f(-x)))=f(x)f(-x). \end{cases} I have found only that for ...
2
votes
2answers
46 views

Determine function such that $f\left(\sqrt{x_1^2+(f(x_1))^2}\right) = f\left(\sqrt{x_2^2+(f(x_2))^2}\right)$ for every $x_1,x_2$.

Determine a numerical no constant function $f$ such that for all $x_1$, $x_2$ in its domaine of definition, the equality $$f\left(\sqrt{x_1^2+(f(x_1))^2}\right) = ...
11
votes
0answers
100 views

How to solve non-linear differential equation [duplicate]

How to solve non-linear differential equation $$y'(x) = y(y(x)), \quad y\colon\mathbb{R}\to\mathbb{R}?$$ Of course, $y(x)\not\equiv 0$. If we substitute $y(x) = Ax^n$, we get complex $n$ and $A$. Any ...
31
votes
4answers
979 views

Is it true that this function $f(n)=n^{13}$?

Assume strictly monotone increasing function; such that $f:N^{+}\to N^{+}$, $h$ for all $n\in N^{+}$, $$f(f(f(n)))=f(f(n))\cdot f(n)\cdot n^{2015}$$ Prove or disprove:$f(n)=n^{13}$ ...
6
votes
1answer
62 views

Is there a constructive discontinuous exponential function? [duplicate]

It is well-known that the only continuous functions $f\colon\mathbb R\to\mathbb R^+$ satisfying $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbb R$ are the familiar exponential functions. (Prove ...
3
votes
1answer
44 views

Functional equation $f(f(y))+f(x-y)=f(xf(y)-x)$

Find all functions $f$ defined on the real and takes real values ​​such that $$f(f(y))+f(x-y)=f(xf(y)-x)$$ for all $x,y\in \mathbb{R} $ My approach: Let be x=y=o, so $f(f(0))+f(0)=f(0) \to ...
6
votes
1answer
124 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
5
votes
2answers
255 views

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y \in \mathbb{R}^n$

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying \begin{equation*} f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n. \end{equation*} My attempt: I manage to show ...
0
votes
1answer
28 views

Recurrence equation $f(g\cdot x)/f(x)=t$

I am trying to solve equation $\frac{f(g x)}{f(x)}=t$ and I'm out of ideas. Any suggestions? In my problem, $f$ is a continuous and weakly increasing function.
0
votes
2answers
40 views

The functional equation $x(x+1)+C(x)=(x+1)(x+2)+C(x+1)=(x+2)(x+3)+C(x+2)=…$

Consider the functional equation $$x(x+1)+C(x)=(x+1)(x+2)+C(x+1)=(x+2)(x+3)+C(x+2)=...$$ The equality continues to infinity. Is there $C(x)$ that satisfies all the equality? If there is, what is it? ...
2
votes
1answer
79 views

An awkward Functional Equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x+1))^2$$ for all $x,y \in \mathbb{R}.$ I proved that $f$ is bijective, but I am stuck there. Any help please?
1
vote
1answer
110 views

Functional equation $f(x)=f(x-1)+f(x-a)$

Please help to solve functional equation for real numbers We have: $a\in \mathbb{R}$ and $a>1$ $f_a(x)=1$ if $x<a$ $f_a(x)=f_a(x-1)+f_a(x-a)$ for $x\ge a$ @update Actually we have to find ...
3
votes
1answer
33 views

D'alembert functional equation

The D'Alembert functional equation is $f(x+y)+f(x-y)=2f(x)f(y)$. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfy the functional equation for all $x,y\in\mathbb{R}$. It's well known that $f$ is of the ...
2
votes
2answers
83 views

All solutions to functional equation $f(x+1)-f(x)=1$

I was thinking of the possibility of finding all solutions other than $f(x)=x$ for the functional equation: $f(x+1)-f(x)=1$ If there are other solutions, what will be some restrictions for the ...
7
votes
6answers
206 views

Solve the following equation: $\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {64x + 5}}}} - \sqrt x= 1$

A past examination paper had the following question that I found interesting. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? $$\sqrt {x + ...
0
votes
0answers
21 views

Stability of Cauchy exponential functional equation

If f : R → R is a function satisfying |f(x + y)- a^xy f(x) f(y)| ≤ δ for all x,y ∈ R and for some positive δ, where a is a positive real constant, then show that either the function $f(x) ...
1
vote
1answer
37 views

Integral equation involving Planck radiation formula

I am stuck in solving the following integral equation: $$\sigma T^4=\pi\int_{\lambda_0}^{\lambda_1}d\lambda W_{\lambda,T}$$ where: ...
8
votes
1answer
90 views

Complicated real to real functional equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x)+1)^2$$ for all $x,y \in \mathbb{R}.$ So far I have proved that $f$ is bijective. How should I continue?
4
votes
1answer
51 views

A functional equation with no term outside functions

Find all $f:\mathbb{N} \rightarrow \mathbb{N}$ satisfying $$f(m-n+f(n))=f(m)+f(n)$$ for all $m,n \in \mathbb{N}.$ I have no idea about how to find them, because there are no terms outside of the ...
4
votes
1answer
52 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
3
votes
3answers
199 views

What function satisfies $f(x)+f(−x)=f(x^2)$?

What function satisfies $f(x)+f(−x)=f(x^2)$? $f(x)=0$ is obviously a solution to the above functional equation. We can assume f is continuous or differentiable or similar (if needed).
0
votes
1answer
60 views

Trigonometric functional equations. I need hints for this problem

Find all functions $ f,g : \mathbb R \to \mathbb R $ that satisfy the functional equation $$g(x)f(y) = f \left(\frac{x + y}{2}\right)^2-f\left(\frac{x-y}{2}\right)^2$$ for all $x,y \in \mathbb R$.
0
votes
2answers
82 views

Solving the functional equation $f(x + y) + g(x-y) = \lambda g(x) f(y)$

Let $\lambda$ be a nonzero real constant. Find all functions $f,g : \mathbb R \rightarrow \mathbb R$ that satisfy the functional equation for all $x,y \in\Bbb R$: $$f(x + y) + g(x-y) = \lambda ...
2
votes
1answer
78 views

Proof regarding a probability generating function (Poisson)

Let $f(s)$ be the probability generating function ($pgf$) of a non-negative, integer valued random variable. It is also given that $f(1-p+ps)f(p) = f(ps)$. Prove that $f(s) = e^{\lambda(s-1)}$ for ...
1
vote
0answers
47 views

Cauchy's function

An exercise in Real and Complex analysis (exercise 18, page 195,chapter 9) Show that if a function $f$ on the real line $\mathbb{R}$ satisfies \begin{equation*} f(x+y)=f(x)+f(y) ...
0
votes
1answer
33 views

Pexerized Dalembert funtional equation..

Let $\lambda$ be a nonzero real constant. Find all functions $f,g: \Bbb R \rightarrow \Bbb R$ that satisfy the functional equation $f(x+y)+g(x−y)=\lambda f(x)g(y)$. I try this : Let $y=0$ in the ...
2
votes
4answers
73 views

Solve: $f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x).f(\frac{1}{y})$

Here i have one functional equation: If $$f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x)\cdot f(\tfrac{1}{y})\text{ for all x},y\in\mathbb{R}-{0}$$ and $f(0) = \frac{1}{2}$ , then find the value of ...
2
votes
2answers
80 views

How do I prove this function is monotonic?

Let $f:\mathbb R\to \mathbb R$ be a function such that $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$ for every $x,y\in \mathbb R$ and $f(1)=1$. In order to prove this function is 1-1, I just need to prove ...
0
votes
0answers
30 views

D'Alembert's functional equation. I need to solve this problem

Let $λ$ be a nonzero real constant. Find all functions $f,g \colon \mathbb{R} \to \mathbb{R}$ that satisfy the functional equation $$ f(x + y) + g(x-y) = \lambda f(x) g(y) $$ for all $x,y \in ...
3
votes
2answers
113 views

Functional equation $f'(x)=cf(x+1)$ has a solution if and only if $c\leq 1/e$

In Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991 by Gabor J Szekely, problem F.57 there is the study of $f~:~[0,\infty)\to (0,\infty)$ such that: $\exists c>0, \forall ...
1
vote
0answers
31 views

Solving a particular Functional Differential Equation

Suppose we have the following functional differential equation: $$f(a_0+a_1x+a_2f(x))(b_1+b_2f'(x))=c_0+c_1x+c_2f(x)$$ It is easy to see that a linear function: $f(x)=d_0+d_1x$, with appropriate ...
1
vote
1answer
62 views

Prove equation using taylor series

Given $f(x)$, knowing that $f'(x)$ and $f''(x)$ exist for every $0\leq x\leq1$, and provided I know that $f(0)=f(1)$ and that for each $0\leq x\leq1$, $|f''(x)|\leq A$, how can I prove that ...
0
votes
5answers
79 views

Suppose $f$ is a real function satisfying $f(x+f(x))$ = $4f(x)$ and $f(1) = 4$. Then the value of $f(21)$?

Should I proceed with just putting the value of $f(1)=4$ in the first equation or there will be a different way of solving this ?
-1
votes
1answer
60 views

Find all functions $f: \mathbb{Z} \to \mathbb{Z} $ such that for all $x,y, \in \mathbb{Z}$, $f(x-y+f(y))=f(x)+f(y)$. [closed]

Need help proving the following: Find all functions $f: \mathbb{Z} \to \mathbb{Z} $ such that for all $x,y, \in \mathbb{Z}$, $f(x-y+f(y))=f(x)+f(y)$.
2
votes
2answers
38 views

Functional equation $f(f(x))=2x$ on $\mathbb{Z}_{>0}$

I have this functional equation: $$f(f(x))=2x$$ with $f: \mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$. And I want to know if it is possible to list all solutions. I already know that ...
3
votes
5answers
126 views

$f(x)f(1/x)=f(x)+f(1/x)$

Find a function $f(x)$ such that: $$f(x)f(1/x)=f(x)+f(1/x)$$ with $f(4)=65$. I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$ which leaves $f(1/x)$ as: ...
1
vote
1answer
64 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
2
votes
1answer
37 views

Solutions of $f : \Bbb Q^*_+ \to \Bbb Q$ such as $f(\tfrac1x)=f(x)$ and $f(x) = (1+\tfrac1x)f(x-1)$ for all $x$

This is an extension to : Functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ What can be said about functions $f : \Bbb Q^*_+ \to \Bbb Q$ such as ...
3
votes
1answer
67 views

Functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$

How to find all functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ for $x> 1$?
4
votes
1answer
79 views

Solving the functional equation $f(x^2+f(y))=(f(x))^2+y$

Find all $ f : R\rightarrow R $ such that $f(x^2+f(y))=(f(x))^2+y, \forall\text{ x,y}\in R$ Thanks in advance!
0
votes
0answers
38 views

Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in ...
2
votes
2answers
41 views

Can we solve recurrence relations indexed by R?

Recurrence relations are equations with functions from $\mathbb N\rightarrow \mathbb N$. For example given $a:\mathbb N\rightarrow \mathbb N$ (write $a(n)$ as $a_n$) solving the recurrence relation ...
0
votes
0answers
33 views

The most general solution to the functional equation

Suppose we have the following functional equation $$X(x_1,x_2)Y(x_1,x_3)=Z(x_2,x_3).$$ Is the most general solution given by $$X(x_1,x_2)=A(x_1)B(x_2), Y(x_1,x_3)=\frac{C(x_3)}{A(x_1)}, ...
0
votes
1answer
20 views

How can one solve this equation in $Z^2$?!

Ho can one solve the egality $2x+3y=xy$ ? I have to find a value of $x$ in fonction of $y$ so ? I have to add somthing and substrate it I added -2xy then $2x(1+y)-3y(1+x)=0$ Here im suck Can ...
1
vote
2answers
24 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
1
vote
2answers
43 views

Deterministic condition for the nature of one real root of a cubic equation

A cubic equation $ax^3+bx^2+cx+d=0 \space$ where, $a\neq 0$ always has one real root. Is there any direct condition for determining the nature i.e. sign of one real root for sure? Is it possible ...