Let \(L=L\left(x, D_{x}ight)=\sum_{i j} a_{i j}(x) \partial_{i} \partial_{j}+\sum_{j} b_{j}(x) \partial_{j}+c(x)\) be a second order differential operator. Determine its formal adjoint \(L^{*}\left(y, D_{y}ight)\), i.e. the linear operator satisfying \(\langle L u, \phiangle_{L^{2}}=\left\langle u, L^{*} \phiightangle_{L^{2}}\) for all \(u, \phi \in \mathcal{C}_{c}^{\infty}\left(\mathbb{R}^{d}ight)\). What is the formal adjoint of the operator \(L(x, D)+\frac{\partial}{\partial t}\) ?