Find the function $f(x)$ if $f(x+y)=f(x-y)+2f(y)+xy$

I think you made no mistake. It's just that the conditions are incoherent, they offer no solution (even without the condition $f'(0)=1$). There is no differentiable $f$ that satisfies $$\tag1 f(x+y)=f(x-y)+2f(y)+xy. $$ If we take $(1)$ and differentiate with respect to $x$, we get $$\tag2 f'(x+y)=f'(x-y)+y. $$ If instead we differentiate $(1)$ with respect to $y$, we get $$\tag3 f'(x+y)=-f'(x-y)+2f'(y)+x. $$ If we add $(2)$ and $(3)$ we get $$ 2f'(x+y)=2f'(y)+x+y. $$ When $x=0$, we get $$ 2f'(y)=2f'(y)+y, $$ so $(1)$ can only hold for a differentiable function if $y=0$.