Find all functions satisfying the functional equation $ xf(x) + f(1-x) = x^3 - x $

Question

Find all functions, for all real x, that satisfy the following functional equation:

$$ xf(x) + f(1-x) = x^3 - x $$

Answer 1

We have \begin{align*} xf(x) + f(1-x) &= x^3-x\\ (1-x)f(1-x) + f(1-(1-x)) &= (1-x)^3 - (1-x) \end{align*} and hence \begin{align*} f(x) + (1-x)f(1-x) = (1-x)^3 - (1-x) \end{align*} Multiplying the first equation by $1-x$ and subtracting from the third equation, we get \begin{align*} (x(1-x) - 1)f(x) &= (x^3-x)(1-x) - (1-x)^3 + (1-x)\\ (-x^2+x-1)f(x) &= x^3-x - x^4 + x^2 - (1-3x+3x^2-x^3) + 1-x \\ &= -x^4 + 2x^3-2x^2 +x \end{align*} Hence \begin{align*} f(x) = \frac{-x^4 + 2x^3-2x^2 +x }{-x^2+x-1} = x^2 - x = x(x-1) \end{align*}

Answer 2

Plug in $x=1$ and $x=0$ in the given relation to get $f(0)=f(1)=0$ and plug in $x=1/2$ to get $f(1/2)=-1/4$. Apply Lagrange's formula for interpolation:

$f(x)=\frac{(x-1)(x-1/2)}{(0-1)(0-1/2)}.f(0)+\frac{(x-0)(x-1/2)}{(1-0)(1-1/2)}.f(1)+\frac{(x-0)(x-1)}{(1/2-0)(1/2-1)}.f(1/2)=x(x-1)$