<s>[INST] Find $f(x)$ satisfying the functional equation $x^2{f(x)} +f(1- x) =2x -x^4$ A function $f(x)$ satisfies the  functional equation $x^2{f(x)} +f(1- x) =2x -x^4$ for all real $x$. Then $f(x)$ is given by.
My work 
$$x^2{f(x)} +f(1- x) =2x -x^4$$
Replacing $x$ $by$ $1- x$ 
$$(1-x)^2{f(1 - x)} +f(x) =2x -x^4 + 4x^3 -6x^2 + 1$$
Subtracting them
$${f(x)}(1-x^2) +{f(1- x)}((1-x)^2 - 1) = 4x^3 -6x^2 + 1$$
What should I do next ?
 [/INST] It is easier to solve a much more general problem:
Consider the functional equation
a(x) f(x) + b(x) f (1-x) = g(x).
Call a^* (x) = a(1-x) etc. and the equation becomes the two equations.
a f + b f^* = g
a^* f^* + b^* f = g^*. which is a 2-by-2 linear system:
Solving for f by inverting a 2-by-2 matrix gives
f(x) = (1/ (a a^* - b b^)) ( a^ g - b^* g^*).
This works for all functions a,b,g as long as a a^* - b b^* is not zero.
The given problem has a(x) = x^2, b(x) = 1, and g(x) = 2x-x^4.
Plugging these in gives
f(x) = (1/ (x^2(1-x)^2 -1)) ( (1-x)^2 g(x) - g(1-x) )
 = (1/ (x^2(1-x)^2 -1)) ( (1-x)^2 (2x-x^4) - 2(1-x) + (1-x)^4) 

which simplifies to 1-x^2.
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