Matematické Fórum

stuart clark · 24. 12. 2011 14:48

find all continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ which satisfying $3f(2x+1)=f(x)+5x$

ruamaixanh

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stuart clark · 29. 12. 2011 08:58

↑ ruamaixanh:

thanks

ruamaixanh · 29. 12. 2011 19:34

You're welcome, but I'm sorry I made a mistake $h^n(x)$ is not $0$ but $-1$ , which is still a concrete value.

vanok · 30. 12. 2011 04:47 — Editoval vanok (30. 12. 2011 04:49)

Hi ↑ stuart clark:,
Just an observation:
the solutions of the form $\varphi(x)= ax +b$ verify
$3$a(2x+1)+b $=ax+b+5x$
What gives us: $6ax +3a +3b=(a+5)x+b$
and finally $a=1$ and $b= -\frac 32$
Thus we have the following polynomial solution
$\varphi(x)= x -\frac 32$

Let us look now for solution of the form $F( x) +\varphi(x)$
It give us one new equation
$3F(2x+1) =F(x)$

I do not still know if she admits of the other one solutions than the zero solution.

Sincerely Vanok