Problem in solving functional equation $f(x^2 + yf(x)) = xf(x+y)$

Question

To find all functions $f$ which is a real function from $\Bbb R \to \Bbb R$ satisfying the relation $$f(x^2 + yf(x)) = xf(x+y)$$ It can be easily seen that the identity function $i.e.$ $f(x)=x$ and $f(x)=0$ (verified just now) satisfies the above relation!! And putting $y=0$ I have got $f(x^2)=xf(x)$.

Help needed to find other functions satisfying the relation.

Answer 1

Putting $x=y=0$: $$f(0)=0$$

Putting $y=-x$: $$ f(x^2-xf(x))=0$$

If there exists $a\neq 0$ such that $f(a)=0$, then put $x=a$: $$f(a^2)=af(a+y)$$

Then $f(x)$ is a constant, and easily find that $f(x)=0$.

If $f(x)=0$ iff $x=0$, then $x^2-xf(x)=0$ for all $x$, then $f(x)=x$ for all $x$.

Answer 2

Hint: Note that the relation is true for all $x,y$. Therefore

$$f(x^2) = xf(x)$$

this can be solved as:

$$f(x) = C x$$.

Note also that

$$f(x^2 -xf(x)) = xf(0) $$

Answer 3

Substitute $x=y=0$, we have $f(0)=0$.

Suppose $f(a)=0$ for some $a\ne 0$, then substitute $x=a$, gives $$f(a^2)=af(a+y) \hspace{1cm}\forall y$$ Hence we have the trivial solution $$f(y)=\text{constant}=f(a)=f(0)=0$$ for all $y$.

Therefore, if other solutions exist, they must satisfy $f(x)\ne0$ for all $x\ne0$.

Now for $x\ne 0$ let $$y=-x^2/f(x)$$ then $$f(0)=xf(x-x^2/(f(x)))=0$$

Hence we have $$x-x^2/f(x)=0$$ which implies $$f(x)=x$$ for all $x\ne 0$.

Combined with $f(0)=0$, the non-trivial solution is $$f(x)=x$$ for all $x$.

Answer 4

An incomplete solution. We present two results on this problem.

We show that if $f(x)$ is injective then the only non-trivial solution to the functional equation is $f(x)=x$.

Proof: Apart from the trivial solution $f(x)=0$ let us assume $f(x)$ is not identically zero and it is injective. Set $x=y=0$ to get $$f(0^2+0f(0))=0f(0)\Rightarrow f(0)=0$$ Then let $x=-y$ one would obtain $$f(x^2-xf(x))=xf(x-x)=xf(0)\Rightarrow f(x^2-xf(x))=0$$ Using the assumption that $f(x)$ is injective then $$x^2-xf(x)=0\Rightarrow f(x)=x$$

$f(x)$ is an odd function.

Set $y=0$ then $$f(x^2)=xf(x)$$ on the other hand $$f((-x)^2)=-xf(-x)\Rightarrow f(x^2)=-xf(-x)\Rightarrow f(x)=-f(-x)$$ and hence $f(x)$ would be an odd function.