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Don't solve the problem, only pointers so I can solve it!

I have an assignment in which I need to determine two linearly independent solutions $f$, s.t. $x^2f=0$. Secondly four solutions under same requisites, s.t. $x^2f''=0$. For $f\in D'(\mathbb{R})$, i.e. distribution in the dual of $C_0^\infty(\mathbb{R})$.

I've determined one solution in either case, since it's obvious for $\delta_0$. However I'm unsure on how to produce such functions/distributions in general.

Additionally my textbook is Distributions and Operators by Gerd Grubb, so any direction given for it are appreciated.

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    while it is a safe assumption, you might want to state the domain in question. – 2017-02-27
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    $x^2 f(x) = 0$ implies that $f$ is zero for $x\not= 0$. The only distributions that has support at a single point are $\delta$ and it's derivatives (see e.g. [this](http://www-users.math.umn.edu/~Garrett/m/fun/distns_at_zero.pdf) for a proof). This helps to solve the first problem. For the second one you can for example first solve $x^2g(x) = 0$ and then $f''(x) = g(x)$. – 2017-02-27
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    I'm not sure how to flag this question as answered, but thanks! – 2017-02-28

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