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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0answers
41 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
0
votes
0answers
7 views

Rotate an implicit surface

Say I have a the implicit equation: $F(x,y,z)=(x^2+y^2+z^2+R^2-r^2)^2-4R^2(x^2+y^2)$ for $R>r>0$. Which gives me a torus laying on the XY plane. How can I modify the equation that it might lie ...
0
votes
2answers
42 views

Equation going infinitely towards y = 10

I'm programming a site on which I sell services, and the more the customers spends, the more discount they will have. Please have a look at the diagram below. Spending 700 USD will result in 5% ...
0
votes
3answers
41 views

Is there anyway to find the values for equation with two variables

Is there mathematical anyway to find the values for equation with two variables? as example: I have y = (20x+3)r+x and y is 50. then what is x and r values?
0
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1answer
34 views

How can I find the R values

I have the following equations : $$\begin{cases}K = \frac{B – 3}{20}\\ K = (20S+3)R+S\\ K = 20S^2 + (20N+7)S + N\\ N= R - S \end{cases}$$ - And I have the $B$ values, e.g : 834343, 3253538, ...
1
vote
1answer
20 views

Changing the length scale of the system of coordinates

Change the length scale on the axes of original system of coordinates, in which the equation $$y=x^3-px\qquad\text{(1)}$$ is plotted, i.e. introduce new coordinates $x_1$ and $y_1$ instead of ...
0
votes
1answer
53 views

how can I find equation variables?

I have the following equations : $$\begin{cases}K = \frac{B – 3}{20}\\ K = (20S+3)R+S\\ K = 20S^2 + (20N+7)S + N\\ N=S-R \end{cases}$$ - And I have the $B$ values, e.g : 173, 283, 2343, 834343 ...
0
votes
0answers
11 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
0
votes
3answers
36 views

How to write a summation function that counts the number of nodes in a tree?

I come from a programming background and am interested in learning how to represent some things as simple equations, as an entry into thinking mathematically. How do you represent a tree structure as ...
5
votes
1answer
87 views

Solve for $f(x)$ if $f(f(x))=6x-f(x)$

If $f: [0,\infty) \rightarrow [0,\infty)$ and $f(f(x))=6x-f(x)$ $f(x)>0$ $ \forall x \in (0,\infty) $ Find f(x)
6
votes
4answers
79 views

Continuity of function where $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function which satisfies $f(x+y) = f(x)f(y) ~~\forall x, y \in \mathbb{R}$ is continuous at $x=0$, then it is continuous at every point of $\mathbb{R}$. ...
8
votes
1answer
113 views

If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$

Question Let the function $f:\mathbb R\to\mathbb R$,and such $$f(xf(y)+x)=xy+f(x)$$ Find all $f(x)$ Let $x=1,y=1$,then $$f(f(1)+1)=1+f(1)$$ let $f(1)=t$,then $$f(t+1)=1+t$$ So I guess ...
1
vote
2answers
61 views

$f$ is a linear function. If $f(x+3) + f(4x+5) = 10x$,$ f(x)=?$

Question: Suppose $f$ is a linear function. If $f(x+3) + f(4x+5) = 10x$, then $f(x)=?$ My attempts: Okay, this is what I have so far... $$f(x) + f(3) + 5(4x) + f(5)$$ $$f(5x) = 10x - f(8)$$ ...
2
votes
0answers
44 views

If $f(2x-f(x))=x$ . Find all bijective functions.

It is given that $f :[0,1] \rightarrow [0,1] $ and it is bijective. If $f(2x-f(x))=x$ , find all such f. Is my solution correct? My attempt $f(x)$ is bijective. thus there exists g(x) which is the ...
5
votes
0answers
191 views
+100

First order differential equation involving inverse function

I am wondering if there is a way to solve a differential equation of the following form: $$\displaystyle \frac{f'(x)}{x} = \frac{1}{f^{-1}(x)} + \frac{1}{k}$$ We can assume that $f(x): [0,T] \to ...
6
votes
2answers
110 views

if the function $f(f(n))+f(n)=2n+2014$,find the $f$

let the function $f:N^{+}\to N^{+}$,and such $$f(f(n))+f(n)=2n+2014$$ Find the $f(n)$ My try: let $n=1$,then we have $$f(f(1))+f(1)=2016$$ let $f(1)=a$,then $$f(a)+a=2016$$ and let ...
0
votes
0answers
21 views

Existence of a function satisfying all the given infima

Given a function $f : \mathbb{R} \to \mathbb{R}$, we can compute its infimum over $A$ for all the Borel measurable $A \subset \mathbb{R}$. I am wondering when we can deduce in the other direction, ...
2
votes
0answers
27 views

Correct math notation to show min and max of a list of numbers and sign?

I am working on a paper that uses a calculation from DIN 15018. The calculation is only described in text and not as a math equation. I would like to display as an actual equation if possible. The ...
0
votes
1answer
22 views

Re-expressing a function

Is it possible to re-express the function $$ f(t+x_1,t+x_2,x_1,x_2)=x_1+x_2+t $$ as $f(y_1,y_2,y_3,y_4)=???$
2
votes
2answers
131 views

Solutions to functional equation $f(x_1,x_2)+g(a_1t+x_1,a_2t+x_2)+h(b_1t+x_1,b_2t+x_2) = t^2$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$, $g:\mathbb{R}^2 \rightarrow \mathbb{R}$ and $h:\mathbb{R}^2 \rightarrow \mathbb{R}$ be twice continuously differentiable functions such that, \begin{align*} ...
1
vote
2answers
70 views

Identifying the exponential function $f(x)=e^x$ from its functional equation

Prove that if $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(x)=1+xg(x)$ where $\lim_{x\to 0}g(x)=1$, then: a) $\exists f'(x)$ $\forall x$ b) $f(x)=e^x$ I would really appreciate your help.
5
votes
1answer
70 views

Expressing a Recursion in terms of factorials

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = \Gamma(n+1)$$ ...
2
votes
1answer
59 views

Can polynomials with degree at least 2 over $\mathbb{R}$ have finite number of solutions in $End(\mathbb{R},+)$

Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian ...
4
votes
5answers
147 views

Solutions to functional equation $f(f(x))=x$

Is there any more solutions to this functional equation $f(f(x))=x$? I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.
0
votes
1answer
15 views

Able to adjust a Function/Formula with Weighted Variables to correct for changes in a variable?

I apologize for the somewhat cryptic title as I don't quite know how to word it. I have a somewhat abstract question that may have a simple answer. But I am wracking my mind all over this! So ...
2
votes
1answer
54 views

Solutions to functional equation $f(at+x)+g(x)+h(t,bt+x) =0 $

Let $a \neq 0$ and $b \neq 0$ be fixed constants with $a \neq b$. Find all twice continuously differentiable functions $f:\mathbb{R}\rightarrow \mathbb{R}$, $g:\mathbb{R}\rightarrow \mathbb{R}$ and ...
1
vote
2answers
44 views

Functional equation with strange property about irrational numbers

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number r, and any real number x we have $f(x)=f(x+r)$. Show that f is a constant function. It's easy to see any constant ...
2
votes
1answer
58 views

Functional equation with cyclic function.

Find all functions $f:\mathbb R \to \mathbb R$ that satisfy: $$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x.$$ Some progress: I plugged-in $\dfrac{x-1}{x}$ and $\dfrac{1}{1-x}$, got a system of ...
1
vote
1answer
22 views

Uniqueness of Pexider's functional equation

Let $f:\mathbb{R}\rightarrow\mathbb{R}$, $g:\mathbb{R}\rightarrow\mathbb{R}$, and $h:\mathbb{R}\rightarrow\mathbb{R}$ and consider Pexider's equation, $$ f(x) + g(y) = h(x + y) \qquad \qquad (1) $$ ...
3
votes
0answers
26 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
0
votes
0answers
13 views

obtain the continuous form of a master equation

I have the following master equation: $$ \partial_tP(x,y,t)=-[x+(1-x-y)]P(x,y,t)\\ +(x+\epsilon)P(x+\epsilon,y-\epsilon,t)\\ +(1-x-y+\epsilon)P(x,y-\epsilon,t) $$ where $\epsilon$ is a small number, ...
17
votes
3answers
1k views

I am searching for an unusual real-valued function.

It needs to fulfill following condition for all positive $x,y ∈ ℝ$: $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ any ideas?
1
vote
0answers
14 views

Need a equation that defines a certain number

Im programming a function but I just cant structure the equation. I think this is the right place to ask since the problem is completely mathematics. Let me explain three scenarios. There are 4 ...
0
votes
0answers
81 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
1
vote
1answer
44 views

Extending $f(p^k)$ where $p$ is prime

If we have a function $f(x)$, for which we know that $f(p^k)=(p^s+1)^k p^{sk}$ where $p$ is prime, $k$ is a real number, and $s$ is a constant, how do we find $f(x)$? My try: let $k=\log_p(x)$, so ...
5
votes
2answers
168 views

Help with complicated functional equation

Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that: $$f(p,q,r)=\\ =\begin{cases} 0, & \text{ if } pqr = 0 \\ 1 + ...
0
votes
2answers
26 views

Equation Theory: A polynomial with specific remainders when divided by specific divisors. What is the remainder when divided by BOTH divisors

Again, for my Equation Theory class, I have the subject question.$p(x)$ has a remainder of 3 when divided by $x-1$ and a remainder of 5 when divided by $x-3$. What is the remainder when $p(x)$ is ...
4
votes
4answers
88 views

Polynomials that satisfy $(x-1)(p(x+1))=(x+2)(p(x))$ where $p(2)=12$?

I am taking a graduate class on Equation Theory and one of my homework questions asks me to "Determine all polynomials $p(x)$ such that $(x-1)(p(x+1))=(x+2)(p(x))$ and $p(2)=12$. A provided hint is to ...
9
votes
2answers
96 views

Functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$

I was looking for examples of real valued functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$. Preferably, I'd like them to be continuous, differentiable, etc. Of course, there are the constant functions ...
6
votes
0answers
89 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
0
votes
0answers
27 views

An equation with a nested function

I'm trying to find the function $\eta(x)$ such that $\eta(x) F(\eta(x))-G(\eta(x)) = \eta(x)H(x) - G(x)$ but I have no idea how to go about it, or where to look. Thanks for the inputs. All ...
0
votes
0answers
43 views

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$?

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$ for all $y\in \mathbb{R}$? Under what conditions would this imply that $f(x,y)=g(x,y)$?
0
votes
2answers
40 views

What functions satisfy the condition $f(x,y)=g(x)$?

Are there any functions $f(x,y)$ and $g(x)$ that satisfy (1) $f(x,y)=g(x)$ for all $x \in \mathbb{R}$ and $y\in \mathbb{R}$ (2) $f(x,y)$ is not constant in $y$ for each $x$ (i.e. for each $x$ there ...
0
votes
0answers
30 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
1
vote
0answers
39 views

General Solution to Linear Difference Equation: Is this correct

Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + ...
2
votes
0answers
37 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
0
votes
0answers
18 views

Methods of showing that a non-trivial solution exists for a functional equation?

I am just looking at a one variable functional equation, I won't put it down here because it is university related, and I keep thinking that 0 is the only possible solution so it got me wondering how ...
1
vote
1answer
28 views

Solving a second level functional equation over all functions $g$

I am trying to find a closed form expression $f$ such that $$f(g(x+1) - g(x)) + f(g(x) - g(x-1)) = f(g(x))$$ For all functions $g$ I have concluded that for polynomials $$2^{n+1}f(0) = f(a_0 + ...
3
votes
3answers
114 views

Solve functional equation $f(f(f(x)))+f(x)=2x$

Please help me solve this functional equation: find $f(x)$ given that $$f(f(f(x)))+f(x)=2x$$ Thanks very much.
2
votes
1answer
72 views

Find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(mx+c)=mf(x)+c$

Find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(mx+c)=mf(x)+c$, $m≠1$. I know that $f(x)=x$ and $f(x)=c/(1-m)$ are two solutions. But to completely solve it I have no idea. Can we ...