 Solution

Choose $M<\infty$ such that $|X|,|Y|\le M$ almost surely. Equality of the positive moments, together with equality of the zeroth moments $\mathbb{E}[1]=1$, givesfor every polynomial $p$. By the Weierstrass approximation theorem, every continuous function on $[-M,M]$ is a uniform limit of polynomials. Expectations preserve this limit, so $X$ and $Y$ have equal expectations for every continuous function on their common compact support. The moment determinacy on a compact interval therefore gives equality of their laws.