Question

Find all $f:\mathbb{R}_+\to\mathbb{R}_+$ such that $(x+y) \, f\big(f(x) \, y\big)=x^2 f\big(f(x)+f(y)\big)$ for all $x,y>0$.

score 769 · Answer 1 · 2018-06-29 17:59:45

769

Views

Find all $f:\mathbb{R}_+\to\mathbb{R}_+$ such that $(x+y) \, f\big(f(x) \, y\big)=x^2 f\big(f(x)+f(y)\big)$ for all $x,y>0$.

Published on 29 Jun 2018 - 17:59

score 47 · Answer 2 · 2026-03-25 11:06:31

47

Views

If a complicated function $F$ satisfies $F(x,y)=R(x,y)$, where $R$ is a rational function, can we safely take limits on both sides?

Published on 25 Mar 2026 - 11:06

score 76 · Answer 3 · 2018-07-02 17:34:09

76

Views

Functional equation $f(x)=\frac 1T \sum_{t=1}^Tf(a+tx)$

Published on 02 Jul 2018 - 17:34

score 51 · Answer 4 · 2018-07-03 10:49:51

51

Views

Solving PDE where boundary conditions explicitly expressed in equation

Published on 03 Jul 2018 - 10:49

score 823 · Answer 5 · 2026-03-25 06:03:18

823

Views

Solving the infinite radical $\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+...}}}}$

Published on 25 Mar 2026 - 6:03

score 37 · Answer 6 · 2018-07-04 22:12:43

37

Views

Find $f,$ given that $\sum_{i=1}^n f (p_i (x_1,...,x_k))=0 $

Published on 04 Jul 2018 - 22:12

score 169 · Answer 7 · 2018-07-05 06:13:02

169

Views

$f:\mathbb Z^+\to\mathbb Z^+$ satisfying $f(n+1)\in\{4f(n)-1,f(n)-1\}$, with a unique $n\in\mathbb Z^+$ such that $f(n)=m$ for all $m\in\mathbb Z^+$

Published on 05 Jul 2018 - 6:13

score 96 · Answer 8 · 2018-07-05 06:59:23

96

Views

all functions that uphold $f'(x)=f(1/x)$

Published on 05 Jul 2018 - 6:59

score 84 · Answer 9 · 2018-07-05 09:38:31

84

Views

Summing combinations with repetition

Published on 05 Jul 2018 - 9:38

score 82 · Answer 10 · 2018-07-06 11:23:45

82

Views

About $ f(x) + c \space f(g(x)) = h(x) $

Published on 06 Jul 2018 - 11:23

score 136 · Answer 11 · 2018-07-07 08:15:01

136

Views

If $\int^{x}_{0}2x(f(t))^2dt = \bigg(\int^{x}_{0}2f(x-t)dt\bigg)^2$ and $f(1) = 1,$ Then $f(x)$ is

Published on 07 Jul 2018 - 8:15

score 114 · Answer 12 · 2026-02-23 01:08:25

114

Views

$0\leq f,g\in C^{1},\int_{a}^{b}\sqrt{f}\geq\int_{a}^{b}\sqrt{g}\implies\int_a^b f\geq\int_{a}^{b}g$?

Published on 23 Feb 2026 - 1:08

score 55 · Answer 13 · 2018-07-08 18:09:21

55

Views

Is it possible to solve a PDE that depends on explicit evaluations of only 1 parameter, and if so, how?

Published on 08 Jul 2018 - 18:09

score 326 · Answer 14 · 2018-07-10 18:24:30

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $\varphi(f(x+y))=\varphi(f(x))+\varphi(f(y))\quad\forall x,y\in\mathbb{N}$

score 85 · Answer 15 · 2018-07-13 09:40:32

85

Views

How to solve the functional equation $f(x^n+y^n)=f(x^n)+f(y^n)$ for any positive integar $n$?

Published on 13 Jul 2018 - 9:40

List Question

Find all $f:\mathbb{R}_+\to\mathbb{R}_+$ such that $(x+y) \, f\big(f(x) \, y\big)=x^2 f\big(f(x)+f(y)\big)$ for all $x,y>0$.

If a complicated function $F$ satisfies $F(x,y)=R(x,y)$, where $R$ is a rational function, can we safely take limits on both sides?

Functional equation $f(x)=\frac 1T \sum_{t=1}^Tf(a+tx)$

Solving PDE where boundary conditions explicitly expressed in equation

Solving the infinite radical $\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+...}}}}$

Find $f,$ given that $\sum_{i=1}^n f (p_i (x_1,...,x_k))=0 $

$f:\mathbb Z^+\to\mathbb Z^+$ satisfying $f(n+1)\in\{4f(n)-1,f(n)-1\}$, with a unique $n\in\mathbb Z^+$ such that $f(n)=m$ for all $m\in\mathbb Z^+$

all functions that uphold $f'(x)=f(1/x)$

Summing combinations with repetition

About $ f(x) + c \space f(g(x)) = h(x) $

If $\int^{x}_{0}2x(f(t))^2dt = \bigg(\int^{x}_{0}2f(x-t)dt\bigg)^2$ and $f(1) = 1,$ Then $f(x)$ is

$0\leq f,g\in C^{1},\int_{a}^{b}\sqrt{f}\geq\int_{a}^{b}\sqrt{g}\implies\int_a^b f\geq\int_{a}^{b}g$?

Is it possible to solve a PDE that depends on explicit evaluations of only 1 parameter, and if so, how?

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $\varphi(f(x+y))=\varphi(f(x))+\varphi(f(y))\quad\forall x,y\in\mathbb{N}$

How to solve the functional equation $f(x^n+y^n)=f(x^n)+f(y^n)$ for any positive integar $n$?

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