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.