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Let $g$ is a function and $x\in \mathbb{R}$ and $$g(x +2)+g(x-3) = 4x+8$$

How can we find $g(x)$?

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    One such $g(x) =2x+5$ – 2017-02-19
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    We see $g''(x+2)=-g''(x-3)$ so $g''$ is an odd function with period $T=5$. One of well-known case is $\sin''=\sin$ and we let $g(x)=k\sin(\dfrac{\pi}{5}x+\beta)+2x+5$. This function satisfies $g(x +2)+g(x-3) = 4x+8$. – 2017-02-19

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The simplest assumption is $g$ be linear. With $g(x)=ax+b$ we find $a=2$ and $b=5$.

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    How do we know there aren't nonlinear solutions? – 2017-02-19
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    Its better to start with $$g(x)=\sum_{r=0}^na_rx^r$$ and establish that $a_r=0$ for $r\ge2$ – 2017-02-19
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    It's possible it has infinite solutions. – 2017-02-19
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    @labbhattacharjee Why can we assume that $g$ is a polynomial? – 2017-02-19
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    @labbhattacharjee thanks. this assumption leads us to $r=2$ and then $g(x)=ax+b$. – 2017-02-19