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A linear functional equation

Understand the problem

Find all functions $ f : \mathbb R \rightarrow \mathbb R$ such that $ (x + y)(f(x) - f(y)) = (x -y)f(x + y)$ for all $ x, y\in \mathbb R$

Source of the problem
Singapore Team Selection Test 2008
Topic
Functional Equations
Difficulty Level
Medium
Suggested Book
Functional Equations by BJ Venkatachala

Start with hints

Do you really need a hint? Try it first!

Play with choices of x,y belonging to \{0,1,-1\}. Show that f(0)=0.

Play with choices of x,y from the set \left\{-\frac{z}{2},\frac{z}{2}+1,\frac{z}{2}-1\right\}.

Following hint 2, you should be able to get equations that can be added to cancel some terms.

Putting x=\frac{z}{2}-1,y=\frac{z}{2}+1
we get f(\frac{z}{2}-1)-f(\frac{z}{2}+1)=-\frac{2}{z}f(z) Putting z=\frac{z}{2}+1,y=-\frac{z}{2} we get f(\frac{z}{2}+1)-f(-\frac{z}{2})=(z+1)f(1) x=\frac{z}{2},y=\frac{z}{2}-1 gives f(-\frac{z}{2})-f(\frac{z}{2}-1)=(z-1)f(-1)

Adding, we get f(z)=\frac{f(1)+f(-1)}{2}z^2+\frac{f(1)-f(-1)}{2}z.

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