Complete coverage of analytic solution methods could begin with a quick look at separation of variables in subsection 1.3.3, then work methodically through the first five sections of chapter 4: 4.1, Basic Definitions; 4.2, Separation of Variables; 4.3, Characteristic Equations; 4.4, Undetermined Coefficients; and 4.5, Variations of Parameters. Section 4.6, Uniqueness and Existence, might reasonably be part of such a thorough treatment.
Sections 6.5, Analyzing Models without Forcing, and 6.7, Analyzing Models with Forcing, contain examples of the application and utility of characteristic equations and undetermined coefficients (e.g., to the discovery of resonance). In the context of a thorough study of analytic methods, section 6.8, Linear versus Nonlinear, shows how constant-coefficient linear equations, those for which students have solution methods, arise naturally in the course of a linear stability analysis.
The four sections of chapter 8 emphasize constant-coefficient systems: 8.1, Basic Definitions: Systems; 8.2, Constant-coefficient Homogeneous Systems; 8.3, Connections with the Phase Plane; 8.4, Nonhomogeneous Systems: Variation of Parameters. Section 8.3 plays a role parallel to that of section 6.8, Linear versus Nonlinear, for second-order equations. It connects nonlinear and linear systems through a linearized analysis in the phase plane, illustrating the importance of the eigenvalue-eigenvector understanding of constant-coefficient, homogeneous systems. Of course, the phase plane perspective of section 8.3 is much richer than the elementary analytic view taken in section 6.8.
Note that the vector-matrix view of systems is introduced gradually in section 8.1, Basic Definitions: Systems, without assuming prior instruction in linear algebra. Additional review of matrix concepts is provided in appendix section A.5, material you may wish to incorporate if your students are particularly uncertain about these ideas.
A syllabus that intended to cover every analytic method remaining in the text would then proceed sequentially from 9.2, Boundary-value Problems, through section 9.4, Time-dependent Diffusion (to derive the heat equation), section 9.5, Fourier Methods, chapter 10, The Laplace Transform, and chapter 11, More Analytic Methods for Two Dimensions.
If partial differential equations and Laplace transforms were not of interest, one could move directly from the diffusion models and boundary-value problems of sections 9.1-9.2 to sections 11.1-11.3 to cover reduction of order, Cauchy-Euler equations, and variation of parameters, solution tools that can handle such non-constant-coefficient equations as models of diffusion in a circle.
A more ambitious tour of analytic methods would work through section 9.5, Fourier Methods, a long section that should be covered one subsection at a time. It could then turn to sections 11.4 and 11.5 for series solution methods, motivated in part by the eigenfunction equation that arises from separating the heat equation in a circle. Of course, the five sections of chapter 10, The Laplace Transform, could be covered last to complete a full tour in a different order.
In summary, a complete tour of all analytic methods would visit the following sections: