Question

Calculus functional equation: Find all positive $f(x)$ such that $\int f\left(x\right)dx\cdot \int \frac{1}{f\left(x\right)}dx=-1$

score 128 · Answer 1 · 2023-07-17 17:08:25

128

Views

Calculus functional equation: Find all positive $f(x)$ such that $\int f\left(x\right)dx\cdot \int \frac{1}{f\left(x\right)}dx=-1$

Published on 17 Jul 2023 - 17:08

score 73 · Answer 2 · 2023-07-17 18:45:56

73

Views

How to solve the functional equation $f(a+2b)-f(a+1)=f(b)$

Published on 17 Jul 2023 - 18:45

score 165 · Answer 3 · 2026-03-25 21:54:09

165

Views

How to evaluate $\displaystyle \int_{-\pi/2}^{\pi/2} f(x) dx$ where $f(x)=\cos(x)+\sin(f(x))$

Published on 25 Mar 2026 - 21:54

score 65 · Answer 4 · 2023-07-19 19:16:08

65

Views

Find all $f(x)$ such that $f(4x) + 2f(x+1) = \sqrt{x} + \sqrt{x+1}$, for all $x \ge 0$.

Published on 19 Jul 2023 - 19:16

score 132 · Answer 5 · 2026-03-25 21:51:08

132

Views

Find all functions $f∶ \Bbb R \to \Bbb R$ such that, for all $n \in \Bbb N, \left[\frac{{f(x+n) - f(x-n)}}{{2n}} = f'(x)\right]$

Published on 25 Mar 2026 - 21:51

score 136 · Answer 6 · 2023-07-20 06:27:50

136

Views

Solving the functional equation $f(x)=\frac{2-x^2}{2} f(\frac{x^2}{2-x^2})$

Published on 20 Jul 2023 - 6:27

score 360 · Answer 7 · 2023-07-20 14:06:25

360

Views

Solving $f(z) = f( z-z^2)$ for $f : \mathbb{C} \to \mathbb{C}$ analytic

Published on 20 Jul 2023 - 14:06

score 65 · Answer 8 · 2023-07-19 14:24:02

65

Views

2004 Schweitzer problem 4

Published on 19 Jul 2023 - 14:24

score 114 · Answer 9 · 2023-07-21 12:17:22

114

Views

Is there a method to solve the equation $f( f(2x) - f(x) ) = x$?

Published on 21 Jul 2023 - 12:17

score 77 · Answer 10 · 2023-07-21 16:41:13

77

Views

How to solve ODE's given evaluations and constraints of an equation or its derivatives, but not the equations themselves?

Published on 21 Jul 2023 - 16:41

score 184 · Answer 11 · 2026-03-26 22:14:11

184

Views

Shortcut Collatz function satisfies a particular functional equation. Has this approach been studied yet, and if so where are the reference articles?

Published on 26 Mar 2026 - 22:14

score 84 · Answer 12 · 2023-07-22 06:14:29

84

Views

Find all functions over $\mathbb{Z}^+$ satisfying this divisibility criterion

Published on 22 Jul 2023 - 6:14

score 135 · Answer 13 · 2023-07-22 12:26:35

135

Views

Function on the real plane that can be expressed in terms of a function on the real line

Published on 22 Jul 2023 - 12:26

score 160 · Answer 14 · 2023-07-23 15:37:10

160

Views

How to solve this functional equation: $f(x^2) = f(x) f^{-1}(x)$?

Published on 23 Jul 2023 - 15:37

score 40 · Answer 15 · 2023-07-25 16:39:44

40

Views

Existence of solution of functional equation on $S^1$.

Published on 25 Jul 2023 - 16:39

List Question

Calculus functional equation: Find all positive $f(x)$ such that $\int f\left(x\right)dx\cdot \int \frac{1}{f\left(x\right)}dx=-1$

How to solve the functional equation $f(a+2b)-f(a+1)=f(b)$

How to evaluate $\displaystyle \int_{-\pi/2}^{\pi/2} f(x) dx$ where $f(x)=\cos(x)+\sin(f(x))$

Find all $f(x)$ such that $f(4x) + 2f(x+1) = \sqrt{x} + \sqrt{x+1}$, for all $x \ge 0$.

Find all functions $f∶ \Bbb R \to \Bbb R$ such that, for all $n \in \Bbb N, \left[\frac{{f(x+n) - f(x-n)}}{{2n}} = f'(x)\right]$

Solving the functional equation $f(x)=\frac{2-x^2}{2} f(\frac{x^2}{2-x^2})$

Solving $f(z) = f( z-z^2)$ for $f : \mathbb{C} \to \mathbb{C}$ analytic

2004 Schweitzer problem 4

Is there a method to solve the equation $f( f(2x) - f(x) ) = x$?

How to solve ODE's given evaluations and constraints of an equation or its derivatives, but not the equations themselves?

Shortcut Collatz function satisfies a particular functional equation. Has this approach been studied yet, and if so where are the reference articles?

Find all functions over $\mathbb{Z}^+$ satisfying this divisibility criterion

Function on the real plane that can be expressed in terms of a function on the real line

How to solve this functional equation: $f(x^2) = f(x) f^{-1}(x)$?

Existence of solution of functional equation on $S^1$.

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