EDB — 1VG

Exercises

1. [1VG] Note:exercise 4, written exam, June 23th, 2012.

   A function \(f (x) =∑_{n=0}^∞ a_ n x_ n\), analytic in a neighborhood of 0, satisfies on its domain the conditions

   \[ \begin{cases} f ’ (x) = 1 + f (−x)\\ f (0) = c \end{cases} \quad ; \]

   (note that this is not a Cauchy problem!).

   • Determine \(f\).

   • Prove that the function found is the only solution, in the set of all functions that can be derived in a neighborhood of 0.