Conformally flat coordinate system on a surface

Question

Let $f:U\subset \mathbb{R}^2 \to \mathbb{R}$ be a smooth map and Let $S \subset \mathbb{R}^3$ be the graph of $f$. We know that every surface is conformally flat where the metric is given from the Euclidean space $\mathbb{R}^3$. I would like to know the coordinate system $(x,y)$ on S such that its metric is of the form $g=e^{\lambda(x,y)} ((dx)^2+(dy)^2)$ for some function $\lambda$.