EDB — 1VG

Exercises

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

   A function $f(x)={\sum }_{n=0}^{\infty }{a}_n{x}_n$, analytic in a neighborhood of 0, satisfies on its domain the conditions

   $$\{\begin{array}{l}{f}^{\prime }(x)=1+f(-x)\\ f(0)=c\end{array};$$

   (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.