Finding the distribution which solves a given equation

Question

I know how to find a $u$ in the space of distributions on $\mathbb{R},$ i.e. $D'(\mathbb{R})$ which solves $xu=1.$ But I can't seem to be able to find $u$ in $D'(\mathbb{R})$ which solves $(x^2)u=1.$

Thanks for the help in advance!!

Answer 1

Note that you can write $(x^2)u$ as $x(xu),$ so you have the equation $x(xu)=1.$ This has the form $xv=1,$ which has solutions $v=\mathrm{pv}\frac1x + A\delta.$ Thus you get the equation $xu = \mathrm{pv}\frac1x + A\delta.$ This has solutions $u=\mathrm{fp}\frac{1}{x^2}-A\delta'+B\delta.$