Solution 1
We first prove that $f$ is odd .
Note that $f(0) = f(x^2 - x^2) = xf(x) - xf(x) = 0$ , and for nonzero $y$ , $xf(x) + yf(-y) = f(x^2 - y^2) = xf(x) - yf(y)$ , or $yf(-y) = -yf(y)$ , which implies $f(-y) = -f(y)$ . Therefore $f$ is odd. Henceforth, we shall assume that all variables are non-negative.
If we let $y = 0$ , then we obtain $f(x^2) = xf(x)$ . Therefore the problem's condition becomes
.
But for any $a,b$ , we may set $x = \sqrt{a}$ , $y = \sqrt{b}$ to obtain
.
(It is well known that the only continuous solutions to this functional equation are of the form $f(x) = kx$ , but there do exist other solutions to this which are not solutions to the equation of this problem.)
We may let $a = 2t$ , $b = t$ to obtain $2f(t) = f(2t)$ .
Letting $x = t+1$ and $y = t$ in the original condition yields

But we know $f(2t + 1) = f(2t) + f(1) = 2f(t) + f(1)$ , so we have $2f(t) + f(1) = f(t) + tf(1) + f(1)$ , or
.
Hence all solutions to our equation are of the form $f(x) = kx$ . It is easy to see that real value of $k$ will suffice.
Solution 2
As in the first solution, we obtain the result that $f$ satisfies the condition
.
We note that
.
Since $f(2t) = 2f(t)$ , this is equal to

It follows that $f$ must be of the form $f(x) = kx$ .

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.