EDB — 1VC

Exercises

1. [1VC]Difficulty:*.Note:exercise 3, written exam, June 30th, 2017.

   Consider the problem

   \[ \begin{cases} y’(x)=y(x^ 2)\\ y(0)=1 \end{cases} \]

   (this is not a Cauchy problem).

   • Show that, for every \(r {\lt} 1\), there is only one solution defined on \(I = (−r, r)\), and deduce that the same is true for \(r = 1\).

   • Show that the solution is representable as the sum of a power series centered in \(0\) and converging on the interval \([−1, 1]\).

   Solution 1

   [1VD]↺↻