A minimal treatment of analytic solution methods requires a glance at separation of variables (section 1.3.3), basic definitions and techniques for first- and second-order linear, constant-coefficient scalar equations (4.1, 4.3-4.4, 6.1-6.4, 6.6), and constant-coefficient, homogeneous, linear systems (8.1-8.2). Extrapolate between this bare minimum and the full coverage of described in the previous subsection to incorporate additional material you believe appropriate.