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A particular solution is: $f(x) = \frac{1}{2} \sqrt{x}$, obs: don't use this as a given, this is just to show that there is at least one solution.

What method should i use to solve this type of equation?

JaberMac
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    Set $g(x)=f(x)-\frac12\sqrt{x}$. Then you get the homogeneous equation $g(4x)+2g(x+1)=0$. Now it depends on if $f$ is continuous, at least at $x=\frac43$. – Lutz Lehmann Jul 19 '23 at 19:31
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    Using your base case, define $g(x)=f(x)-\frac12\sqrt x$ then you need to solve: $$g(4x)=-2g(x+1)$$ – Thomas Andrews Jul 19 '23 at 19:32
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    Where did you get this question, and are there any assumptions on the regularity of $f$? – Bruno B Jul 19 '23 at 19:43
  • Without further assumptions in $f$ there are infinite solutions, which I think will be hard to describe in a simple way. – jjagmath Jul 20 '23 at 00:18

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