how to solve functional equation?

How to solve this functional equation?

• how do you solve the functional equation f(x)^2=1+x*f(x+1)? I know what the answer is (its f(x)= x+1), but i want to know why. How do you get this?

• Answer:

  I don't know of any known methodical approach. However, you can hope that any solution is from a particularly nice family of functions. If you "guess" that a polynomial solution exists, then for polynomial f of degree n, f(x)² would be of degree 2n and xf(x+1) would be of degree n + 1. This leads to 2n = n + 1 ==> n = 1. So a polynomial solution would have to be a line. Then put f(x) = ax + b. Substitute. (ax + b)² = a²x² + 2abx + b² = 1 + x(ax + a + b) = ax² + (a + b)x + 1. Matching powers, a² = a ==> a = 0 or a = 1 b² = 1 ==> b = -1 or b = 1. And 2ab = a + b. The only solution to all three equations is a = b = 1. So the polynomial solution is f(x) = x + 1 Note that this does not preclude the possibility that there is a solution of a different type.