The following problem arose for an example that I wanted to give in my paper:
Prove (or disprove) that there exists a function $g\colon {\mathbb R_+} \to \mathbb{R}$ such that $$\displaystyle \int_{\mathbb R_+} g(y) x \mathrm{e}^{-yx} \mathrm{d} y = x^2 \text{.} $$ If it exists, state what $g$ is.
As for the motivation for this problem: For example, if the question was finding $g\colon {\mathbb R_+} \to \mathbb{R}$, such that $$\displaystyle \int_{\mathbb R_+} g(y) x \mathrm{e}^{-yx} \mathrm{d} y = \frac1{x^2} $$ The answer would easily be $g(y) = \frac12 y^2 $
Even if this can't be solved, is there a reference that might be useful for solving such problems?