When are two metrics related by a coordinate transformation?

Question

Let there be two metrics $g_{\mu\nu}(x)$ and $\tilde g_{\mu\nu}(x)$. How do I check if there exists a coordinate transformation $y^\mu(x)$ such that $\tilde g_{\mu\nu}(y(x))=\frac{\partial x^\alpha}{\partial y^\mu}\frac{\partial x^\beta}{\partial y^\nu}g_{\alpha\beta}(x)$?

I.e. how can I decide if $g$ and $\tilde g$ are related by a coordinate transformation?