Find all functions satisfying the functional equation $ xf(x) + f(1-x) = x^3 - x $ 
Find all functions, for all real x, that satisfy the following functional equation: 
$$ xf(x) + f(1-x) = x^3 - x $$ 


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*}
