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 ...

learn more… | top users | synonyms

2
votes
1answer
367 views

Equation with a definite integral - can I differentiate it?

I have an equation like this: $$te^{t} = \int_0^t e^\tau u(\tau)d\tau$$ I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how can I do it ...
17
votes
3answers
2k views

Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?

I was wondering if there is a name for a function that satisfies the conditions $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$? Thanks and regards!
1
vote
2answers
164 views

Finding 2^2^2^2^2 … n times

f(1) = 2 f(2) = 2^2 f(3) = 2^(2^2) f(4) = 2^(2^(2^2)) etc how can I find a closed form solution to this? Thanks.
2
votes
1answer
358 views

How to solve algebraically the equation $x = \frac{1}{2}\cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$

How to solve this trigonometric equation $x = \frac 1 2 \cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$ ? The iterative solution seems to be 1.417. Can anybody suggest an algebraic ...
1
vote
1answer
173 views

Functional Equation Analysis

Given: $$F(F(n)) = n$$ $$F(F(n + 2) + 2) = n$$ $$F(0) = 1$$ where n is a non-negative integer. $$F(129) = ?$$ How can we solve such kind of functional equations? Is there any simpler approach ...
3
votes
1answer
161 views

$f(1-f(x))=f(x)$

Find all continuous $f:[0,1] \rightarrow [0,1]$ such that $f(1-f(x))=f(x)$.
4
votes
1answer
430 views

$f(x+f(y))=f(x-f(y))+4xf(y)$

Find all functions $f:R\rightarrow R$ which satisfy $f(x+f(y))=f(x-f(y))+4xf(y)$ $\forall x,y \in R$. I strongly suspect $0$ and $x^2+C$ to be the only solutions but, as is almost the case with ...
14
votes
3answers
758 views

The Notorious Triangle Problem

I was told this question by a friend, who said that their friend had thought about it on and off for six months without any luck. I have then had it for a while without any luck either. It is in the ...
0
votes
1answer
99 views

$x^3[f(x+1)-f(x)]=1$

Possible Duplicate: $x^3[f(x+1)-f(x-1)]=1$ Given that f is continuous and $x^3[f(x+1)-f(x)]=1$, determine $\lim_{x\rightarrow \infty}f(x)$ explicitly.
0
votes
1answer
110 views

$x^3[f(x+1)-f(x-1)]=1$

Given that $x^3[f(x+1)-f(x-1)]=1$, determine $\lim_{x\rightarrow \infty}(f(x)-f(x-1))$ explicitly.
2
votes
3answers
239 views

Solving the functional equation $x[f(x+1)-f(x-1)]=1$ [duplicate]

Possible Duplicate: Solving the functional equation $f(x+1) - f(x-1) = g(x)$ How do I approach this problem $x[f(x+1)-f(x-1)]=1$.
7
votes
4answers
668 views

Solving the functional equation $f(x+1) - f(x-1) = g(x)$

Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies $$ f(x+1) - f(x-1) = g(x) $$
3
votes
0answers
77 views

Function Shape Reference

I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague. I would like to have a ...
9
votes
3answers
417 views

Evaluating $f(x) f(x/2) f(x/4) f(x/8) \cdots$

Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for $$\begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) ...
5
votes
2answers
1k views

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

I was discussing with a friend of mine about her research and I came across this problem. The problem essentially boils down to this. $f(x)$ is a function defined in $[0,1]$ such that $f(x) + f(1-x) ...
8
votes
2answers
765 views

On sort-of-linear functions

Background A function $ f: \mathbb{R}^n \rightarrow \mathbb{R} \ $ is linear if it satisfies $$ (1)\;\; f(x+y) = f(x) + f(y) \ , \ and $$ $$ (2)\;\; f(\alpha x) = \alpha f(x) $$ for all $ x,y \in ...
0
votes
2answers
136 views

How to solve the following system?

I need to find the function c(k), knowing that $$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$ $$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$ $$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$ ...
1
vote
1answer
184 views

conversion of a powerseries $-3x+4x^2-5x^3+\ldots $ into $ -2+\frac 1 x - 0 - \frac 1 {x^3} + \ldots $

This is initially a funny question, because I've found this on old notes but I do not find/recover my own derivation... But then the question is more general. Q1: I considered the function $ ...
4
votes
1answer
428 views

Find a function such that $f(\log(x)) = x \cdot f(x) $

I recently read an article in which the author describes how to find some functions that obey to certain recursion relationships. If we want to find a function that satisfies, for example, $f(x^a) = ...
1
vote
1answer
1k views

Solve the functional equation $f (x+y)=f (x)+f (y)+xy (x+y)$, $f$ continuous at $0$

Find all functions $f:\mathbb{R} \to \mathbb{R}$, that are continuous at the point $x=0$ and satisfy: $$f (x+y)=f (x)+f (y)+xy (x+y) \ \ \forall x,y \in \mathbb{R} $$
4
votes
3answers
531 views

Find all continuous functions $f$ on $\mathbb{R}$ satisfying $f(x)=f(\sin x)$ for all $x$.

Find all continuous functions $f$ on $\mathbb{R}$ satisfying $f(x)=f(\sin x)$ for all $x$.
2
votes
0answers
198 views

How to find a function which satisfies such functional equation?

How to find a function which satisfies: $$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$ where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 ...
1
vote
3answers
127 views

Solving Series of equations

I have the following series of equations ($n+2$ equations $n+2$ variables): \begin{equation*} k_0q_0+\lambda q_0 + c_0 = 0, \\ k_1q_1+\lambda q_1 + c_1 = 0, \\ k_nq_n+\lambda q_n + c_n = 0, \\ ...
0
votes
1answer
194 views

Solving this set of quadratic equations

I have a set of quadratic equations of the form: \begin{equation*} 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0 \\ 2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0 \\ \vdots \\ ...
5
votes
1answer
215 views

Generalization of cos: is this function known?

Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$). Consider a function $f_2$ ...
6
votes
1answer
322 views

Iterated polynomial problem

Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.
3
votes
2answers
320 views

Implicit function $y = e^{(y-1)/x}$

I'd like to know if the function $ y = f(x) : [0,1] \rightarrow [0,1]$ defined implicitly by the transcendental equation $$\displaystyle y = e^{(y-1)/x}$$ is "well known" (name, properties) or is ...
2
votes
0answers
371 views

Solving $f(f(x))=g(x)$ equations [duplicate]

Possible Duplicate: Square root of a function (in the sense of composition) I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known ...
0
votes
1answer
172 views

Finding $\alpha$ such that $f(\alpha(x+y))=f(x)+f(y)$

Problem taken from the link: http://web.mit.edu/rwbarton/Public/func-eq.pdf I am stating the question here For which $\alpha$ does there exists a nonconstant function $f: \mathbb{R} \to \mathbb{R}$ ...
14
votes
4answers
631 views

Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$

In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)) = f(x) + x$$ (which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is ...
1
vote
2answers
574 views

Solving the functional Equation $f(f(x))=f(x)+x$

Let $f$ be continuous on $\mathbb{R}$. Then how to find all continuous functions satisfying $f(f(x))=f(x)+x$
5
votes
1answer
206 views

How to solve DE that relate values of derivatives at different points?

I try to solve for the specific function $f(x) = \frac{2-2a}{x-1} \int_0^{x-1} f(y) dy + af(x-1)$ It looks similar to the function used to find the Renyi's parking constant because it came out from a ...
7
votes
5answers
1k views

Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$

Find all polynomials $p(x)$ such that for all $x$, we have $$(x-16)p(2x)=16(x-1)p(x)$$ I tried working out with replacing $x$ by $\frac{x}{2},\frac{x}{4},\cdots$, to have $p(2x) \to p(0)$ but then ...
7
votes
1answer
638 views

Polynomials which satisfy $p^{2}(x)-1 = p(x^{2}+1)$

Can we find a polynomial $p(x) \in \mathbb{R}$ such that $\text{deg}\ p(x)>1$ and which satisfies $$p^{2}(x)-1=p(x^{2}+1)$$ for all $x \in \mathbb{R}$. This question can be very well identified with ...
15
votes
3answers
785 views

Continuous function satisfying $f^{k}(x)=f(x^k)$

How does one set out to find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ which satisfy $f^{k}(x)=f(x^k)$ , where $k \in \mathbb{N}$? Motivation: Is $\sin(n^k) ≠ (\sin n)^k$ in general?
2
votes
3answers
1k views

Function Satisfying $f(x)=f(2x+1)$

If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function and satisfies $f(x)=f(2x+1)$, then its not to hard to show that $f$ is a constant. My question is suppose $f$ is continuous and it satisfies ...
24
votes
3answers
3k views

Prove that this function is bounded

This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone ...
15
votes
6answers
750 views

Solution to $1-f(x) = f(-x)$

Can we find $f(x)$ given that $1-f(x) = f(-x)$ for all real $x$? I start by rearranging to: $f(-x) + f(x) = 1$. I can find an example such as $f(x) = |x|$ that works for some values of $x$, but not ...