# Tagged Questions

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### Are Exponential and Trigonometric Functions the Only Non-Trivial Solutions to $F'(x)=F(x+a)$?

Are exponential & trigonometric functions the only non-trivial solutions to $F'(x)=F(x+a)$? $F(x)=0$ would be the trivial solution. Then, for $a=0$ (or $a=2\pi i$), we have $F(x)=e^x$, and ...
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### If $f$ is not identically zero and $|f(x)| \le 1$ for $x$ $\in \mathbb{R}$, then also $|g(x)| \le 1$ for $x \in \mathbb{R}$

Suppose that $f$ and $g$ satisfy the equation $f(x+y)+f(x-y)=2f(x)g(y)$, $x$,$y$ $\in \mathbb{R}$. Show that If $f$ is not identically zero and $|f(x)| \le 1$ for $x$ $\in \mathbb{R}$, then also ...
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### A problem about functional equations

We want to find all continuous functions $f:R→R$ that satisfy the equation $f(x^2+1/4)=f(x)$ for all real x. Of course -If I am right- constant functions satisfy the equation mentioned, and as well ...
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### ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
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### General solution of the recurrence equation with real shifts

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
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### 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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### About the space $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$

I am reading Taylor's Partial differential equations III (nonlinear equations) (Section 1 of Chapter 16, Theorem 1.2), and Sogge's Lectures on Non-linear wave equatuions. I notice that in the energy ...
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### 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 ...
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### Solving the differential equation $f'(x)=af(x+b)$

How does one find all the differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ which satisfy the equation $$f'(x)=af(x+b),\quad \text{for}\quad a,b \in \mathbb{R}?$$ I see that functions ...
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### A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
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### 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 ...
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### 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 $f: \mathbb{R} \to \mathbb{R}$ ...
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### 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 $f(f(x))=f(x)+x$
### 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 ...
### 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 ...