J. J. Duistermaat and J. A. C. Kolk's book proposes the following problem:
Prove that the differential equation $$(1-x^2)^2 u' -2xu=(1-x^2)^2$$ has no solution $u\in \mathcal{D}'(I)$ if $I$ is an open interval with $[-1,1]\subset I$.
Since $x\mapsto (1-x^2)^2$ has roots in $I$, I can't use any of the theorems of the book.
I would apreciate some help.