Find all functions \(f(x)\) satisfying: \[ (x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2) \]

 

Find all functions \(f(x)\) satisfying:

\[ (x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2) \]

Solution

Let's  `u=x+y`and `v=x-y` 

Then, `x=(u+v)/2`  and `y=(u-v)/2`

From the equation : `f(x)` if `(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)`

We will get: `vf(u)-uf(v)=(u^2-v^2)uv`

                  

 `f(u)/u-u^2=f(v)/v-v^2`  for all `u,v` different from `0`

Let: `v=1``rightarrowf(u)/u-u^2=f(1)-1`

Therefore:  `f(u)=u^3+au`  for all `u!=0` and `(a=f(1)-a)`

If `x=y=0` `rightarrow2f(0)=0`  Then `f(0)=0`

Hence,   `f(x)=x^3+ax`.