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

6
votes
2answers
287 views

Solving the functional equation $f(x) = f(x / \phi) f(x / \phi^2 - 1)$

I'm trying to find a function $f(x)$ such that the spacing between consecutive roots looks like the infinite Fibonacci word: $$1, \phi^{-1}, 1, 1, \phi^{-1}, 1, \phi^{-1}, 1, 1, \phi^{-1}, 1, 1, ...
2
votes
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 ...
2
votes
2answers
93 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 ...
2
votes
2answers
44 views

function equation with translation of independent variable

The following has come up in some work I'm doing: If $\frac{f(x+a)}{f(a)}=g(x)$ , where $g(x)$ is given and $a \geq 0$ is a constant, what is $f(x)$ ? We can assume that $g(x)>0 ~ \forall x$ . Of ...
0
votes
2answers
28 views

How can I determine the best relationship for 3 variables, given several data points?

What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions. I have ...
0
votes
2answers
56 views

Some functional equations in two variables

I have two questions. i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation $$ |f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|} $$ has a solution ...
-1
votes
2answers
23 views

Unit price equation.

having some trouble with an equation.. 50x + 40y = 375 75x + 70y = 621 I know x=2.82 and y=5.85 (I looked it up in the answers), but I have literally no idea how to get there... Thanks in advance.
2
votes
1answer
62 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 ...
2
votes
1answer
88 views

Functional Equation $f(n) = 2 f(n / f(n) )$

I have the following functional equation: $f(n) = 2 \cdot f\left(\frac{n}{f(n)}\right)$ Under the precondition that $ f(n) = \omega(1) $, monotinic and a initial value $ f(1) = \Theta(1) $ one can ...
2
votes
1answer
104 views

Functional/Differential Equation

In the midst of a calculation too long (and too irrelevant) to describe here, I've been forced to confront the following equation: $$1-p-f(f(p))-f(p)f'(f(p))=0$$ Here $f$ is a differentiable ...
1
vote
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 $(b)$.
1
vote
1answer
47 views

How to solve $2f(x)+xf(\frac{1}{x})-2f(|\sqrt2 \sin(\pi(x+\frac{1}{4}))|) = 4 \cos^2 \frac{\pi x}{2}+x\cos\frac{\pi}{x}$

I saw a question today. $$2f(x)+xf(\frac{1}{x})-2f(|\sqrt2 \sin(\pi(x+\frac{1}{4}))|) = 4 \cos^2 \frac{\pi x}{2}+x\cos\frac{\pi}{x}$$ It had options like this (one or more than one may be correct): ...
1
vote
1answer
26 views

How to state a recurrence equation

I'm working on my homework and I saw this problem: A divide and conquer algorithm $X$ divides any given problem instance into exactly two smaller problem instances and solve them recursively. ...
1
vote
1answer
121 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 ...
1
vote
1answer
25 views

Is $Tf=x_0+\int_0^tf(t)(1-f)\,dt$ a Lipschitz function?

Consider the Banach space $C[0,1]$ with the uniform norm and the operator given by $(T(f))x=x_0+\int_0^xf(t)(1-f(t))\,dt$ for $x\in[0,1]$ and $x_0\geq0$. I want to show that there is a number ...
1
vote
1answer
33 views

Additive & Multiplicative Link

Is there an isolative property about the following multivariable equation: $$f_1(x)g_1(y)=f_2(x)+g_2(y)$$ That is, is this equation rearrangable such that it may be put in the form ...
1
vote
1answer
76 views

Functional equation- solving techniques

I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed ...
1
vote
1answer
69 views

Solution of functional equation

i know the solutions of the well known Cauchy-functional-equation $f(x+y)=f(x)+f(y)$ But what does it change if i have the following form $f(x+g(y))=f(x)+f(g(y))$ ? what can i say about g? ...
1
vote
1answer
29 views

Linear after change of variables

Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, ...
1
vote
1answer
41 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants ...
1
vote
1answer
58 views

What is the family of functions that satisfies

Let $f$ be a continuous function in $\Re$, differentiable (at least one time). Then, which functions can satisfy: $$ \frac {f(x)} {f'(x)}= - \frac {f(x-\frac {f(x)} {f'(x)})}{f'(x-\frac {f(x)} ...
1
vote
1answer
88 views

A fairly difficult differential equation

I was thinking what if I had a differential equation of the form: $$\frac{d^2u}{dx^2} + vu(x) = 0 $$ where $v(y(x))$, that is $y$ is a function of $x$. What are the possible restrictions that I can ...
1
vote
1answer
56 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
1
vote
1answer
133 views

Applications of mathematics to some kinds of sporting strategies

I am a rather newbie maths person. Haven't studied maths in a while and so not sure what things are called was hoping to get some information to push me in the right direction so I know what it is I ...
1
vote
1answer
178 views

Analytic solution of a certain functional equation

Here's my question. Let $b_2$, ..., $b_d \in \mathbb{C}$ ($d$ is an integer greater than 2), and consider the functional equation $$V(z^d)=dz^{d-1} V(z)+(b_2 z^{d-2} + b_3 z^{d-3}+\ldots + b_d)$$ ...
0
votes
1answer
39 views

If $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$, find $f$

Assume $f: (0, \infty) \to \mathbb{R}$ is a continuous function such that for any $x,y > 0$, $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$. Find $f$. I would work with each condition ...
0
votes
1answer
31 views

Functional equations with nowhere differentiable solutions

As an example, the functional equation $f(x+y)=f(x)f(y)$, by declaring that $f$ is continuous and differentiable, we can arrive at the unique solution $f(x)=a^x$, by first showing that ...
0
votes
1answer
40 views

Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that $f(x)=f(x^y)$

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$f(x)=f(x^y)$$ for all $x,y\in\mathbb{N}$. I'm not intrested in the trivial solution $f(x)=k$, where $k\in\mathbb{N}$.
0
votes
1answer
37 views

How to prove that a function is continuous in functional equation?

I was wondering about different methods or properties to prove that a function in a functional equation is continuous or differentiable. Can somebody give me some examples of such problems or methods, ...
0
votes
1answer
44 views

continuous function and functional equation

Let $ g:\mathbb{R} \to \mathbb{R}$ be a continuous function: $g(x) = g(x+1)$. Show that $g$ satisfies the equation: $g(x)=\frac{1}{4} \left(g(\frac{x}{2})+g(\frac{x+1}{2})\right)$ for all $x$. Is it ...
0
votes
1answer
22 views

Cauchy functional equation three variables

If I have function from $R^3$ to $R$ satisfying $f(x_1,x_2,x_3)+f(y_1,y_2,y_3) = f(x_1+y_1,x_1+y_2,x_3+y_3)$ is it necessarily linear? $f(z_1,z_2,z_3) = \lambda _1 z_1+\lambda _2 z_2+\lambda _3 ...
0
votes
1answer
30 views

Find a 3d equation that goes through a series of points?

I have a series of points in 3d space and I need to find an equation that goes through all of them. What would be the best way to do this? Points: (3.7, 0.45, 0.7) (5.2, 0.8, 0.96) (6, 1.04, 1.15) ...
0
votes
1answer
34 views

How to find a 3d equation from a series of points

I have 6 points and I need to find the equation, or an equation, that will go through all of them. How would I go about doing this? The points are as follows. (3.7, 0.45, 0.7) (5.2, 0.8, 0.96) (6, ...
0
votes
1answer
251 views

modelling trough polynomial equation

Set up a polynomial equation that models each problem below.Then solve the equation, and state the answer to each problem. 1.One dimension of a cube is increased by 1 inch to form a rectangular ...
0
votes
1answer
73 views

rectangular paddock, dimensions, maximise area it encloses

Having trouble trying to work out a question which involves finding a function to graph evidence of the correct answer, any advice would be greatly appreciated. I am struggling with part 'b' a lot, ...
0
votes
1answer
41 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
0
votes
1answer
76 views

Find a function $\Phi$ such that $ \Phi(x)^{T}\Phi(y)=\exp(-\|x-y\|^2/(2\sigma^2))$

It's a question from HW: Suppose we have $ \Phi:\mathbb{R}^p \to \mathbb{R}^\infty $ that satisfies: $$ \Phi\left(x\right)^{T}\Phi\left(y\right)=\exp\left(-\frac{\left\Vert x-y\right\Vert ...
0
votes
1answer
29 views

Representation of a Functional Equation.

Let, $f:\mathbb{R}^{J+1}\to\mathbb{R}$, $g:\mathbb{R}^J\to\mathbb{R}$ and consider the following equation: $$f(x,g(x))=g(x).$$ I would like to know if the following statement is true (or find ...
0
votes
1answer
265 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 ...
0
votes
1answer
96 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 ...
0
votes
1answer
39 views

monotonic function. I need to show ots linear

If a real additive function f is monotonic, then it is linear. I need to show that the monotonic function f that satisfies caushy additives functional equation is linear
0
votes
1answer
15 views

Criteria for elevating a rational FE solution to reals

I have a functional equation which I have solved for rational numbers. Now to 'elevate' my solution to reals, I use the given fact that f is continuous. I could have done the same process, if I knew ...
0
votes
1answer
58 views

Difficult Functional Derivative

I've been working on this problem set for a little bit now and I've finally made it to the last question. I'm now left with this monster and I don't quite understand where to proceed. All the ...
0
votes
1answer
36 views

Problem involving distances on a parametrically defined curve

SO the problem is stated below: A curve is defined paramterically by: $$x(t)=a\cosh t$$ and $$y(t)=b\sinh t$$ where a and b are positive constants and $-\infty < t < \infty$ The expression ...
0
votes
1answer
44 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
0
votes
1answer
43 views

Understanding an Equation and how to implement it

A common method for linking language with psychological variables involves counting words belonging to manually-created categories of language. One counts how often words in a given category are used ...
0
votes
1answer
39 views

Functional Equation and totally multiplicative functions

Find all $f:\mathbb{C}\rightarrow\mathbb{C}$ s.t. $\forall a, b\in \mathbb{C}, f(ab) = f(a)f(b)$. I could deduce a lot of things about what happens at the roots of unity, and 0, but I can't find out ...
0
votes
1answer
44 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
0
votes
1answer
88 views

A functional equation for homogeneous functions of degree zero

Let a function $F: \textbf{R}^2_{++} \rightarrow \textbf{R}$ satisfy the relation: if $F(x,y)=F(x',y')$ then $F(x,y)=F(x+x',y+y')$. It is easy to prove that under the additional assumption of ...
0
votes
1answer
54 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...