<s>[INST] Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x). Problem : 
If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x). 
My approach : 
The given equation can be written as $$(x-y)f(x+y) -(x+y)f(x-y) =4xy(x-y)(x+y)$$
$$\Rightarrow \frac{(x-y)f(x+y)}{(x-y)(x+y)} -\frac{(x+y)f(x-y)}{(x-y)(x+y)} =4xy$$ 
$$\Rightarrow \frac{f(x+y)}{x+y} -\frac{f(x-y)}{x-y} =4xy$$
Now we know that $$(x+y)^2 -(x-y)^2 = 4xy$$ 
$\Rightarrow \frac{f(x+y)}{x+y} =(x+y)^2$....(i)  
& $\frac{f(x-y)}{x-y} =  (x-y)^2$...(ii) 
Now putting y =0  in (i) and (ii) we get : 
$\frac{f(x)}{x} =x^2$ $\Rightarrow f(x) =x^3$ 
But the answer is $f(x) =x^3 +kx$ ( where k is any constant ) please clarify this part thanks...
 [/INST] The additional term comes from the relation $(x-y)k(x+y)-(x+y)k(x-y)=0$ for any $k$
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