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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20 views

How to solve Functional Equations

I have a midterm tomorrow and have been able to cover all other topics except this. I don't even have an idea how to start these questions. If someone could give me some tips I would very much ...
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22 views

Banach fixed-point theorem for a recursive functional equation

I was asked to prove that the functional equation $$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ ...
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2answers
42 views

$f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $

Solve Functional Equation$$f: \mathbb{R}_+ \to \mathbb{R}_+; \; f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $$ I'm stuck in the beginning. Any hint will be helpful.
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2answers
60 views

The 'prime logarithm'

Lately I've been thinking about the functional equation $$f(ab) = f(a) + f(b)$$ but not in the usual sense where continuity or differentiability are assumed. It's clear that $f(1) = 0$, by letting $a ...
2
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2answers
108 views

solve equations - find a short piece of a wire

Problem: A piece of wire 20 feet long is cut into two pieces so that the sum of the squares of the lengths of the two pieces is 202 square feet. What is the length, in feet, of the shorter piece of ...
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0answers
10 views

Functional extrema and the Euler-Lagrange equation

For a functional of the form: $$S(q)=\int_{t_{1}}^{t_{2}}L(q,\dot{q})dt$$ where $\dot{q}=\frac{\partial q}{\partial t}$ , one finds that extrema are reached (to first order) for the condition : ...
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4answers
157 views

How to solve $f'(x)=f'(\frac{x}{2})$

How do we solve this given $f'(0)=-1$. It does not look separable. I can integrate both sides but end up with a functional equation with is not helpful.
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1answer
37 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega ...
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1answer
21 views

Checking my understanding of Cauchy's functional equation.

Cauchy's functional equation is given as $$f(x+y)=f(x)+f(y)$$ Wikipedia states that the solution to this functional equation with $x\in\mathbb Q$ is $f(x)=cx$, where $c$ is an "arbitrary ...
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1answer
39 views

Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...
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3answers
104 views

Mean Value theorem functional equation

I need help solving the following functional equation. Suppose $\,f : \mathbb{R} \to \mathbb{R}$ is differentiable and $$f'\Big(\frac{x + y}{2}\Big) = \frac{f(x) - f(y)}{x -y}$$ holds for all $x ...
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1answer
26 views

Solving functional equation $\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)}$

I want to find a functional equation $f(s,x)$ such that $$\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)}$$ If it helps the context I need this in is where $t$ is a member of a set of real number and ...
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0answers
16 views

euation of modular functions

In this article at page four: It says that one easily checks the equation $(1)$. But I can't check that. I already tried to use $\eta(-1/z)=(-iz)^{1/2}\eta(z),~z$ in the upper half plane, but this ...
3
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0answers
56 views

Find the polynomial $p(x)$

A polynomial $p(x)$ gives a remainder of $1$ when divided by $x^{100}$ and a remainder of $2$ when divided by $(x-2)^3$. Evaluate $p(x)$. By the Remainder Theorem, $p(x)$ can be written as ...
4
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1answer
57 views

How to solve $f(x)+f\left(\frac{1}{x}\right)=e^{x+\frac{1}{x}}$

Given that $f:\mathbb{R}_0 \rightarrow \mathbb{R}_0$ find such $f$ that $$f(x)+f\left(\frac{1}{x}\right)=e^{x+\frac{1}{x}}$$ Note that I came up with this question, and personally am not sure ...
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15 views

A functional equation with an inequality

I have an increasing function on $[0,1]$, $p \mapsto \Pi(p)$, that has the following properties. $$\Pi(0) = 1 - \Pi(1) = 0$$ $$\Pi(p) + \Pi(1-p) < 1 \quad \forall{p} \in (0,1)$$ $$\Pi(p) > p ...
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1answer
48 views

Alternate method for finding $f(x)$

Let f be a real valued function $f:(0,\infty)\to(0,\infty)$ such that it satisfies the relation: $$f(xf(y))=x^2\cdot y^a$$ where $a\in\mathfrak{R}$ then find $f(x)$ and the possible values of $a$. ...
2
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1answer
50 views

Solve a functional equation involving integrals

Let $a, b \in (0,1)$ with $a+b \le 1$. Find all continuous functions $f:[0,1] \rightarrow \mathbb{R}$ having: $$ \int_{0}^{x} f(t) dt = \int_{0}^{ax} f(t) dt + \int_{0}^{bx} f(t) dt \tag1 $$ ...
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2answers
70 views

How to show that If $f(x+y)=f(x)f(y)$ then $f(x)\geq 0$ [duplicate]

I would appreciate if somebody could help me with the following problem: Q: Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function and $f(x+y)=f(x)f(y)$. Then $f(x)\geq 0$
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0answers
33 views

The function $\psi (x+2)=1+\sqrt{2\psi (x)-\psi^2 (x)}$

The function $\psi : \mathbb{R}\rightarrow \mathbb{R}$ satisfies the relation: $$\psi (x+2)=1+\sqrt{2\psi (x)-\psi^2 (x)},$$ for all real $x$. What features it has? Place your example of at ...
12
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272 views

About the derivative of a function defined on rational numbers

I have found this problem: Let $f : \mathbb{Q} → \mathbb{R}$ with property: $$|f(x) − f(y)| \le (x − y)^2 \tag1$$ for all $x, y \in \mathbb{Q}$. Prove $f$ is constant. My idea is to consider ...
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682 views

Find functions $f:\mathbb{R}\to[0,1]$ such that: $\lim_{x \to +\infty} \frac{f(x)^2}{f(2 x)}=1$

Find functions $f:\mathbb{R}\to[0,1]$ that satisfy: $$\lim_{x \to +\infty} \frac{f(x)^2}{f(2 x)}=1,$$ $$f'(x)\leq 0 \, \forall x,$$ and $$\lim_{x \to +\infty} f(x)=0.$$ $$\lim_{x \to -\infty} ...
2
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1answer
29 views

Continuous function on compact interval $[a,b]$ with non-negative values

let $f:[a,b]\longrightarrow[0, \infty)$ be a continuous function satisfying the following: $f(\frac{a+x}{2})+f(\frac{2b+a-x}{2})=f(x), \forall x \in [a,b]$. Then the only function that satisfies these ...
5
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0answers
42 views

Proving a Functional Equation is Differentiable [duplicate]

Let $f:(0,\infty )\rightarrow \mathbb{R}$ satisfy $f(xy)=f(x)+f(y)$ and let f be differentiable at x=1. Prove f is differentiable over it's entire domain with derivative $f'(x)=\frac{f'(1)}{x}$ Using ...
3
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1answer
120 views

If $f\circ f\circ f=id$, then $f=id$ [duplicate]

Let $f$ a continuous function on all $\mathbb R$. How can I prove that if $f\circ f\circ f=id$, then $f=id$ ? I really have no idea.
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1answer
29 views

Functional Equations again

Let $f$ be a differentiable function satisfying $f(x+y)=(f(x))^{\cos y}\cdot (f(y))^{\cos x}$ for all $x,y \in R$. $f(0)=1, f'(0)=\ln 2$. If $$\int_{\frac{\pi}{2}}^{\frac{17\pi}{2}} ...
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18 views

Number of distinct values

Question: How many possible values of (a, b, c, d), with a, b, c, d real, are there such that abc = d, bcd = a, cda = b and dab = c? I tried multiplying all the four equations to get: $$(abcd)^2 = ...
1
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1answer
41 views

Cauchy functional inequality

Given a function on a closed interval $f\colon I\subset \mathbb{R}\to \mathbb{R}$ with $$f(x+y) \leq f(x) + f(y).$$ Moreover, I know that $f$ is monotonic increasing continuous on all points except ...
1
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1answer
55 views

Functional Equation: $f(x^2-y^2)=xf(x)-yf(y)$

Let $\mathbb{R}$ be the set of Real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2-y^2)=xf(x)-yf(y)$$ for all pairs of real numbers $x$ and $y$. This is a problem ...
4
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197 views

Does there exist the function $f^2(x)\ge f(x+y)\left(f(x)+y \right) $

Does there exist the function $f:\mathbb R^+\rightarrow \mathbb R^+$, such that $$f^2(x)\ge f(x+y)\left(f(x)+y \right) \forall x,y \in \mathbb R^+$$ My work so far: Assume that a function ...
1
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1answer
32 views

Judicious guess for the solution of differential equation $y''-6y'+9y= t^{3/2} e^{3t}$

$(a)$ Let $L[y]=y''-2r_1y'+r_1^2y.$ Show that $$L[e^{r_1t}v(t)]=e^{r_1t}v''(t).$$ $(b)$Find the general solution of the equation $$y''-6y'+9y= t^{3/2} e^{3t}$$ I have problems only in part $(b)$.
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35 views

What can be said about $f$ in $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$ if $g$ is known?

Suppose you have the equailty $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$, $x,y\in\mathbb{R}$, and $g$ is known. What non-trivial facts about $f$ can be deduced from this? (Assume $f(x,y)\neq ...
8
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2answers
129 views

Find all functions $f(f(f(…(f(x_1,x_2),x_3),…),x_{2016}))=x_1+x_2+…+x_{2016}$

I am trying to solve the functional equation: Find all functions $f:\mathbb R^2\rightarrow \mathbb R$ such that for all $\left \{x_1,x_2,...,x_{2016} \right \}\subset \mathbb R$: ...
2
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2answers
70 views

continuous (on 3, 4 and 5) f is constant, if $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is continuous on 3, 4 and 5, such that $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$. Show that f is constant. I don't know what to do ...
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1answer
57 views

How to solve this functional equation: $f(1-f(x))=1-x^{9}, f(1)=0$

I have managed to guess one solution of this function : $f(x)=1-x^{3}$, but I have no idea how to prove it unique, or get other solutions. If this is not solvable, how can you prove this function ...
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1answer
25 views

If $g:\mathbb{N}\rightarrow \mathbb{R}$ and $g(m+n)+g(m-n)=2g(m)+2g(n)$ what is $g(x)$

Determine all functions $g:\mathbb{N}\rightarrow \mathbb{R}$ such that $g(1)=1$ and $$g(m+n)+g(m-n)=2g(m)+2g(n), \quad \forall m\ge n \in \mathbb{N}$$ Because of the identity $k\cdot (a+b)^2 ...
4
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1answer
54 views

A function $f$ satisfies the condition $f[f(x) - e^x] = e + 1$ for all $x \in \Bbb R$.

Let $f$ be a function such that $f[f(x) - e^x] = e + 1$ for all $x \in \Bbb R$. Find $f(\ln 2)$. I've considered two cases: $f(x) = e^x + c$, where $c$ is constant. Then $f(c) = e^c + c = e + 1$, ...
2
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1answer
57 views

Functional equation.

I'm trying to solve the functional equation $f(x+f(y)) = f(x)-y$ where $f : \mathbb{Z} \to \mathbb{Z}$. What I got so far is: $f$ is injective and $f(0) = 0$. Thanks in advance for your time.
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1answer
33 views

Find all function that satisfy $(f(x) + xy) \cdot f(x - 3y) + (f(y) + xy) \cdot f(3x - y) = (f(x + y))^2$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all real numbers $x$ and $y$,$$(f(x) + xy) \cdot f(x - 3y) + (f(y) + xy) \cdot f(3x - y) = (f(x + y))^2.$$
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2answers
81 views

Solution of functional equation $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$

If $x,y\in \mathbb{R}$ and $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$ and $f'(0)=0\;,$ Then $f(x)$ is $\bf{My\; Try::}$ Using $$f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow ...
1
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0answers
14 views

For which $a, b, $ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx $ essentially determine $f(x) $

For which values of $a, b, $ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx $ essentially determine $f(x) $. This is a generalization of Solve ...
13
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3answers
691 views

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
0
votes
0answers
16 views

Uniqueness of a solution to a functional equation

I have two complex-valued functions, $f$ and $g$, that satisfy the following properties. $\overline{x}$ denotes the complex conjugate of $x$ below. $$g(t)\overline{g(t+h)} = f(h) \quad ...
5
votes
1answer
110 views

If $f(x-f(y))=f(-x)+(f(y)-2x)\cdot f(-y)$ what is $f(x)$

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x-f(y))=f(-x)+(f(y)-2x)\cdot f(-y), \quad \forall x,y \in \mathbb{R}$$ It's easy to see that $f(x)=x^2$ is a function ...
0
votes
0answers
33 views

Simple almot linear functional equation

I'd like to solve functional equation: $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$ I've managed to get: $f(0)=0,f(n)=0$ for all $n\in N$; $f(\frac{1}{2})=0$; $f(-x)=-f(x)$. I'll be grateful for any help.
0
votes
1answer
20 views

Functional equation on unit square

Suppose $F$ is a continuous function defined on the unit square $[0,1]\times[0,1]$ satisfying the following properties : i) $ F(a,a)=0,$ for all $a\in[0,1],$ ii) $F(a,b)=-F(b,a)$ for all ...
0
votes
1answer
23 views

The functional equation and differentiability

Find all functions $f: \mathbb R\rightarrow \mathbb R$, at the same time satisfying the following two conditions: a) $f (x + yf (x)) = f (x) f (y)$ b) the function $f$ can be represented ...
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vote
0answers
20 views

Find all functions such that $f(m)+f(n)|m^p+n^p$

For fixed prime number $p$, find all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)+f(n)\mid m^p+n^p$ for all $m,n\in \mathbb{Z}^+$ I managed to get only that for prime $q$ we have ...
3
votes
5answers
88 views

Solve the equation $1 / \cos x = \cos x + \sin x$

I'm having trouble solving the equation $$ \frac{1}{\cos x} = \cos x + \sin x $$ For what I understand I have to make the equation $= 0$ So I get $$ \frac{1}{\cos x} - \cos x -\sin x = 0 $$ Any ...
0
votes
0answers
13 views

How to Compute Discrete Intrinsic Curvature

Given a function $f$, a region $S$ where this function is twice differentiable, and the property that all of its discrete difference series centered in $S$ converge within $S$ We have that $$ ...