The Functional Equation $ ( x + y ) \big( f ( x ) - f ( y ) \big) = ( x - y ) f ( x + y ) $, need solution have answer Find all functions $ f: \mathbb R \to \mathbb R $ such that for all reals $ x $ and $ y $,
$$ ( x + y ) \big( f ( x ) - f ( y ) \big) = ( x - y ) f ( x + y ) \text . $$
I actually got the answer by guessing and checking, $ f ( x ) = a x ^ 2 + b x $, but I want to see the solution. My friend suggested surjectivity but I don't see how to continue.
$ f ( x ) = f \left( \frac x { f ( x ) ^ 2 } \right) $;
this is what I just got.
The answer is correct, just need a solution. Thanks!

For any real numbers $ a $ and $ b $, if we define the function $ g : \mathbb R \to \mathbb R $ with $ g ( x ) = f ( x ) - a x ^ 2 - b x $, then using the original functional equation, we'll have
$$ ( x + y ) \big( g ( x ) - g ( y ) \big) = ( x - y ) g ( x + y ) \text . \tag 0 \label 0 $$
In particular, setting $ a = \frac { f ( 1 ) + f ( - 1 ) } 2 $ and $ b = \frac { f ( 1 ) - f ( - 1 ) } 2 $, we will also have $ g ( 1 ) = g ( - 1 ) = 0 $. Letting $ y = 1 $ in \eqref{0}, we get
$$ ( x + 1 ) g ( x ) = ( x - 1 ) g ( x + 1 ) \text , \tag 1 \label 1 $$
while substituting $ x + 1 $ for $ x $ and $ - 1 $ for $ y $ in \eqref{0} we have
$$ x g ( x + 1 ) = ( x + 2 ) g ( x ) \text . \tag 2 \label 2 $$
\eqref{1} and \eqref{2} together show that
$$ 2 g ( x ) = x ( x + 1 ) g ( x ) - ( x - 1 ) ( x + 2 ) g ( x ) = x ( x - 1 ) g ( x + 1 ) - ( x - 1 ) x g ( x + 1 ) = 0 \text . $$
Therefore $ g $ is the constant zero function, and hence $ f ( x ) = a x ^ 2 + b x $ for all $ x \in \mathbb R $. It's straightforward to see that any function of this form satisfies the original functional equation, and thus they form the class of all the solutions.
