The chapters devoted primarily to analytic methods are chapter 4 (first-order scalar), chapter 6 (second-order, mostly constant-coefficient), chapter 8 (systems, mostly constant-coefficient), chapter 10 (Laplace transforms), and chapter 11 (other second-order methods). Boundary-value problems are solved in section 9.2, and Fourier methods for the heat equation are developed in section 9.5, where second-order eigenvalue problems appear as well. Besides teaching the methods themselves, this coverage is organized to illustrate the process of generalization in mathematics by proceeding from simpler to more complex problems.