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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2
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2answers
20 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 ...
0
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0answers
7 views

Functional equation in distribution

Neagu(1984) reformulated the Pompeiu functional equation (PE) in distribution and determined the distributional solution of (PE) My question is what is meaning of "functional equation (PE) in ...
-5
votes
1answer
26 views

Algebra question and solving in elimination equation [on hold]

Logan organized a dog show that raises funds for animal shelter. Logan charges five dollars for each student ticket and eight dollars for each ticket. His total ticket sales were 3585 and the total ...
2
votes
5answers
102 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
46 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
33 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
62 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
73 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
30 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
36 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
30 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
18 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
23 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
32 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 ...
0
votes
1answer
38 views

How to solve a equation with special conditions?

I have this equation: $z = 11n + 13m$ Conditions: $z < 2015$ $z$, $n$ and $m$ must be natural numbers ($>0$). How many options are possible for $z$?
0
votes
1answer
51 views

Find the function satisfying the given condition

If $f(xy)=e^{xy-x-y}[e^y f(x) +e^x f(y)]$ and $f'(1)=e$. $f'$ denotes the derivative of function $f(x)$. Find $f(x)$. I could find that $f(0)=0$ and $f(1)=0$ and then found the derivative the got ...
1
vote
0answers
45 views

Find all differentiable functions $f$ such that $f(f(x))=f'(x)$ [duplicate]

Here is a problem I made up: Find all differentiable functions $f$ from the reals to the reals such that $f(f(x))=f'(x)$ for all real $x$.
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votes
0answers
15 views

Approach on solving limit equation systems and finding some f given assymptotes?

This is a "reverse" question of finding the asymptote of a function Recently, I am interested in doing some sort of modelling which involve equations of the form $$@(t)=1-f(t)$$ where $f(t)$ is ...
4
votes
2answers
151 views

Solve differential equation $f'(z) = e^{-2} (f(z/e))^2$

I'm curious if there's a simple closed solution to the following DE and, if so, what it is. $$\begin{align} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align}$$
2
votes
0answers
73 views

We all know about compositions of functions, but what about decomposition. Is there a way with math, not just heuristics?

The composition operator is a well know and quite often used method in integration and differentiation, think u-substitution. However, given a composition like $$f(f(f(...f(x)...)))$$ Where there are ...
1
vote
1answer
41 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
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votes
0answers
12 views

Iterative Functional Equation? (discrete, increment, logarithm)

Say $f(n)$ is defined as discrete iterative process, $n \in [0,1,2,\ldots,N]$, and $g(x)$, $x \in [-\infty,+\infty]$ is "small": $|g(x)|<1, \forall x$: $$f(n+1)=f(n) + \log(1 + g(f(n)))$$ ...
2
votes
1answer
43 views

Polar to cartesian equation conversion

I have a polar equation defined as: $r = ae^{θ tan (m)}$ where, $a$ and $m$ are constants, θ is the angle between the horizontal axis from the origin (xc,yc) to the coordinate. $e$ refers to ...
5
votes
1answer
85 views

Find $f(x) $ given that: $f'(x)=\frac{f(x)-x}{f(x)+x}$ [closed]

I would appreciate if somebody could help me with the following problem: Find $f(x)$ given that: $f \colon \mathbb{R^+} \rightarrow \mathbb{R^+}$, $f$ is differentiable function, and ...
7
votes
4answers
164 views

$f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous , and $f(x+1)+f(x)=x^2$

I would appreciate if somebody could help me with the following problem: Find $f(x)$ ($f(x)$ is not Polynomial function), given that: $f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is ...
4
votes
0answers
86 views

Integer functional equation $f(f(f(n)))=f(n+1)+1$

Can you find all functions $f:\mathbb N\rightarrow\mathbb N$ satisfying the functional equation $$ f(f(f(n)))=f(n+1)+1 $$
2
votes
0answers
84 views

Solving functional equation for all real numbers.

The functional equation to be solved is $ f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$. Domain:Reals,Codomain:Reals.I found about 4 possible solutions to the equation but ran into a fundamental problem with all ...
1
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0answers
53 views

Iterative (functional) roots of integer functions (functions on $\mathbb{Z}$)

A function $g:A\to A$ is called a $k$-th iterative root of another function $f:A\to A$ ($A$ an arbitrary set and $k\in\mathbb{N}$) iff $f=g^k$, where $g^k(x)=g\circ g\circ\ldots\circ g(x)=g(g(\ldots ...
0
votes
0answers
21 views

Functional equation similar to Babbage's equation

I'm interested in the potential solutions $f: R_{+} \rightarrow R_{+}$ to the functional equation: $ \forall x \in R_{+}, \quad f(s(x) - f(x)) = ax $ where $a>0$ is a constant. $s: R_{+} ...
1
vote
1answer
18 views

Extension of the additive Cauchy functional equation

Let $f\colon (0,\alpha)\to \def\R{\mathbf R}\R$ satisfy $f(x + y)=f(x)+f(y)$ for all $x,y,x + y \in (0,\alpha)$, where $\alpha$ is a positive real number. Show that there exists an additive function ...
1
vote
1answer
28 views

hyperplane in $L^2$

Consider $R^2$ valued functions $f,g \in L^2([0,1],\mu, R^2)$ where $f=(f_1,f_2)$,$g=(g_1,g_2)$ and $\sqrt{\langle f, g\rangle}=\int_{[0,1]} (f_1(x)g_1(x)+f_2(x)g_2(x)) d\mu(x)$ Suppose for a given ...
2
votes
1answer
68 views

What are all pairs of functions f and g so that $f(x)f(y) = g(x+y)$?

It can be shown, and is a problem in Rudin's Principles of Mathematical Analysis (Chapter 8), that when $f$ is continuous, and $f(x)f(y) = f(x+y)$, $f$ is a function of the form $e^{cx}$. Must this ...
2
votes
3answers
66 views

Solving for $f(2004)$ in a given functional equation

Given that $$f(1)=2005$$ and $$f(1)+f(2)+...f(n) = n^{2}f(n)$$ for all $n>1$. Determine the value of $f(2004)$. My progress: I first substituted $n-1$ into the equation to get ...
2
votes
1answer
42 views

A function $\psi(z)$ satisfies the following functional equation…

I was given the following question to solve: "Given a function $\psi_0(z)$ satisfies the following functional equation: $\psi_0(z+1)=\frac{1}{z}+\psi_0(z)$ Prove that ...
0
votes
5answers
54 views

Boolean Simplification of AB + A'+B'

Is there any way to simplify this function? Or is this the simplest equation? : AB + A'+B'
0
votes
0answers
53 views

Extension of the Cauchy functional equations

Let $f:(0,a)\to\mathbb{R}$ satisfying $$f(x+y)=f(x)+f(y)$$ for all $x,y,x+y\in(0,a)$, where $a$ is a positive real number. Show that there exists an additive function $A:\mathbb{R}\to\mathbb{R}$ such ...
2
votes
1answer
34 views

A problem on solving functional equations

If $f(x),\forall x\in\mathbb{R}$ is continous and differentiable, and satisfies: $f(x_1+x_2)+f(x_1-x_2)=2f(x_1)f(x_2),\forall x_1,x_2\in\mathbb{R}$ $f\left(1\right)=\dfrac{3}{2}$ How to prove: ...
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0answers
42 views

Proving existence and uniqueness of solutions to the functional equation $f(n) = r \cdot f(n-1)$

Suppose I have a functional equation $f(n) = r \cdot f(n-1)$ where $r$ is a constant. This represents a geometric progression and a known solution is $g(n) = ar^n$ where $a = g(0)$. By intuition, ...
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votes
0answers
14 views

Solving a linear functional equation

Working with Green functions, I have found to solve the following equation $$ -\omega^2G(\omega)-m^2G(\omega)+\kappa\sum_{n=-\infty}^\infty b_nG(\omega-n\omega_0)=1 $$ where $m$, $\kappa$ and ...
5
votes
2answers
170 views

$f(\alpha x) = f(x)^{\beta}$ under different constraints

With $\alpha > 0,\, \beta \in \Bbb R^*,\, \alpha, \beta \neq 1$ and $f : \Bbb R \to \Bbb R_+^*$, let's consider the functional equation $$ f(\alpha x) = f(x)^{\beta} \tag{$\Xi$}$$ or equivalently ...
-1
votes
1answer
57 views

Follow-up to $f(x)^2 = f(\sqrt2 x)$

This is a follow-up to: Solving $(f(x))^2 = f(\sqrt{2}x)$ . So $f : \Bbb R \to \Bbb R$ is $\mathcal C^2$ and verifies $\forall x,\, f(x)^2 = f(\sqrt2 x)$. We already know that $f(0) \in \{0,1\}$ and ...
10
votes
3answers
254 views

Solving $(f(x))^2 = f(\sqrt{2}x)$

I would like to know how to solve this equation : $$f(x)^2 = f(\sqrt{2}x)$$ We assume that $f : \mathbb R \to \mathbb R$ is $\mathcal C^{2}$. The answer should be $f(x)=e^{-x^{2}/2}$, but I don't ...
1
vote
1answer
53 views

Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $\frac{f(x+3)}{3+f(x)}=\frac{4+x^2}{x^2}$

Find all continuous $f:\mathbb{R}\to\mathbb{R}$ satisfying $$\frac{f(x+3)}{3+f(x)}=\frac{4+x^2}{x^2}.$$ I believe the original question was $$\frac{f(x)}{3+f(x)}=\frac{4+x^2}{x^2},$$ which has a ...
1
vote
1answer
21 views

How to solve the delay algebraic equation $xf(x) + \alpha f(x - {x_0}) - \alpha f(x + {x_0}) = 0$?

In the process of solving a problem, I am faced with the problem of finding a non-zero function $f:\mathbb{R} \to \mathbb{R}$ which satisfies the equation $$xf(x) + \alpha f(x - {x_0}) - \alpha f(x + ...
14
votes
1answer
1k views

Prove that function is continuous without knowing the function explicitly

Let $f\colon \mathbb R^+\to\mathbb R$ be a function that satisfies the following conditions: $$\tag1 \lim_{x\to 1}f(x)=0 $$ $$\tag2f(x_1)+f(x_2)=f(x_1x_2)$$ Show that $f$ is continuous in its domain. ...
1
vote
0answers
19 views

Calculus of Variations for discrete functionals

Question: How does one determine the optimal function f that either maximize or minimizes: $$\int_{x_1}^{x_2} L\left(x, f, D_{h,x}[f ]\right) dx$$ Whereas: $$ D_{h,x}[f] = \frac{f(x + h) - ...
4
votes
0answers
77 views

$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
0
votes
0answers
31 views

Does this problem have analytic/approximate analytic solution?

Define $F[i]=F[i-1]-F[i-1]^c$ with $F[0]=r>0,c\in(0,1)$. Define $R(r,c)=\min\{i:F[i]<1\}$. What is solution to $c\in(0,1)$ in $R(r,c)=r^{1/k}$ with some given $k>1$?
0
votes
0answers
24 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
0
votes
1answer
15 views

Defining a rectangular prism using a formula and complex numbers.

I recently read that a line can be defined using the formula $$ A = O + dL $$ where $A$ represents any point on the line, $O$ represents the vector origin of the line, $L$ represents the direction ...