I am a total beginner at functional equations; I am an undergrad studying them for my Putnam seminar, so I don't really know/understand what I am doing. I just got some results from symbolic manipulations and I am trying to interpret them correctly. If possible, a more heuristic answer would be appreciated. I got the question from Brilliant:
$f:\mathbb{R} \to \mathbb{R}$
$5f(x+y)+y^5=f(x)+(x+y)^5$
I plugged in $x=y=0$, and concluded that $f(0)=0$
I set $y=0$, and I got $f(x) = \dfrac {x^5}{4} \tag 1$
I set $x=0$, and I got $f(y) = 0 \tag 2$
I plugged $(1)$ into the original functional equation, and of course it only worked when $y$ was $0$.
I think from $(2)$ we can conclude that $0$ is the only solution. I guess it's because I think we can maybe come up with more functions such as $(1)$ which satisfy the original equation under some nieche conditions, but once we get $f$ is $0$ for all values in its domain, that feels like the end of the line. However I can't really come up with a cohesive argument.