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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50 views

Can this Delay Differential Equation (DDE) be solved in closed form?

$$\frac{\mathrm{d}\psi}{\mathrm{d}x}=\frac{(x+\mu)^2-\lambda^2}{\mu}\psi(x+\mu)$$ where $\mu\geq0$, $\lambda\in\mathbb{C}$ are constants and $\psi(x)\to0$ as $|x|\to\infty$.
3
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106 views

Weird ordinary differential equation

I am searching for the functions $f,g:[0,\infty) \to [0,\infty)$ which are both increasing, $f'$ is strictly increasing, $g$ is the inverse of $f$ and $$ g(x)^2=f(x)^2\cdot g'(x)$$ My approach was ...
3
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229 views

A Functional Differential Equation

I was having a play with some trig. identities and noticed the following: $\cos{x}=\frac{\sin{2x}}{2\sin{x}}$ Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential ...
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47 views

Soultion of a particular functional differential equation

I need to solve the following functional differential equation: $(4\mu-\lambda r - \lambda x+\lambda)f'(x) = \lambda( f(x) - f(1-x-r))$, where $x\in (0, 1-r)$, $r\in (\frac{1}{2}, 1)$ Or, a more ...
2
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1answer
27 views

Characterizing functions which satisfy de Moivre's theorem

Let $f$ and $g$ be two non-zero functions $ \mathbb R \to \mathbb R $ which are continuous and differentiable everywhere. Furthermore, say that for all integer $n$: $$ (f(\theta) + i g(\theta))^n = f(...
2
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1answer
111 views

Another functional equation: $f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor$

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
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1answer
90 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
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1answer
105 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 ...
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42 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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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): ...
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1answer
27 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. ...
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1answer
122 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 $...
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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 $0<L&...
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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 $\Omega_1(x)=\...
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1answer
77 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 ...
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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? ...
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1answer
30 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, ...
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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 $\lambda,\...
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60 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)} {f'(x)}...
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90 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 ...
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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) = \varphi^{-1}[1+\varphi(x)]...
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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 ...
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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 $$V(z^d)=dz^{d-1} V(z)+(b_2 z^{d-2} + b_3 z^{d-3}+\ldots + b_d)$$ ...
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1answer
17 views

Modify position with variable ratio

I have a line that represents an axis, on the computer's screen, it has a model domain and a screen domain. For example the model domain can be [2, 10], and the screen domain [50, 400]. I need to ...
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17 views

finding proper coefficient for the two graphs to intersect at one point only

We have two functions such as $y=\ln(x)$ and $y=cx^{1/2}$ and I look for the proper positive coefficient $c$ which satisfies that the graphs of the functions above intersects at only one point. If we ...
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1answer
29 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 $m$...
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41 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
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1answer
39 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 $f'(x)=f(0)f(x)$...
0
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1answer
42 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 $k\in\mathbb{N}$.
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38 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, ...
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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
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1answer
23 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? $f(z_1,z_2,z_3) = \lambda _1 z_1+\lambda _2 z_2+\lambda _3 z_3$...
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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) (7....
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35 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, 1....
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261 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 ...
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89 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, ...
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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 ...
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77 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 ^{2}}{2\...
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1answer
30 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 ...
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282 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 ...
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1answer
98 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 ...
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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
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16 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 ...
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62 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 previous ...
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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 ...
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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 ...
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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
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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 ...
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1answer
45 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, $Y_0(t)=\mathcal{L}^{-1}(...
0
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1answer
90 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 ...