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The question is to solve the functional equation $$f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$$ $\forall x,y \in \mathbb R$

I tried to put x=y=0,y=x and y=-x in the given functional equation.I ended up getting $$f(2x)=3f(x)+f(-x)$$ and $$f(x)+f(-x)=x\sqrt{f(x})$$ from where I am unable to proceed.Thanks.

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    At least we know that $f(x)\equiv 0$ is a solution:) – 2017-01-21
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    Another solution is $x^2/4$ – 2017-01-21
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    @LeBtz I said $x^2$ divided by $4$ not just $x^2$ – 2017-01-21
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    @kingW3 Yes, my bad. It still doesnt hold for $x=-1, y=1$ – 2017-01-21
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    Since this question was closed even though the answers in the linked question aswell come to the solution $x^2/4$ (which is wrong), I posted my answer in the linked question again and deleted it here. – 2017-01-22

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