Functional equations in one variable..

Question

How do you solve the functional equation involving only one variable...what if and if not given that $f(x)$ is a polynomial...

Say for example $f(x)=f(x-1) +2x$

Answer 1

This is a finite difference equation of the first order, and it is linear.

Hence you solve the homogeneous part of the equation,

$$f(x)=f(x-1)$$ which is simply $f(x)=c$.

Next you find a particular solution of the non-homogeneous equation. As the RHS is a linear polynomial, you know that a quadratic polynomial will do and

$$ax^2+bx+c=a(x-1)^2+b(x-1)+c+2x$$ gives $a=b=1$.

Hence

$$f(x)=x^2+x+c$$ which can be written $$f(x)=x^2+x+f(0).$$

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Technical note:

If no other condition is given on $f$, the values of $f$ are only constrained one unit apart and the constant can differ when the fractional parts differ. Hence the general solution is actually

$$f(x)=x^2+x+f(\{ x\})$$

where $f$ is any function in $[0,1)$.

An example solution: