Chapter 4, Analytic Tools for One Dimensions, surveys some of the basic solution methods for scalar first-order equations. It treats characteristic equations, undetermined coefficients, and variation of parameters, all to set the stage for subsequent extensions of these methods.
The characteristic equation method is extended to second-order scalar equations in sections 6.3-6.4 (real and complex characteristic roots, respectively) and to systems of first-order equations in section 8.2.
The method of undetermined coefficients is extended to second-order equations in section 6.6. To alter the pattern of generalization, variation of parameters is extended to first-order systems in section 8.4, then specialized to second-order equations in 11.3.
The balance among the analytic methods favors the simpler constant-coefficient methods because they reveal relatively easily most of the features of the behavior of solutions of linear equations. Constant-coefficient equations arise in many models, and they are the natural outcome of linearized analyses. In addition, the inescapable patterns of generalization (e.g., extending characteristic equations from scalar to systems) and specialization (e.g., variation of parameters from systems to second-order) reveal a side of mathematics that too few students at this level appreciate.
Systems of two first-order equations are the primary focus of chapter 8, Analytic Tools for Higher Dimensions. But the geometric (eigenvector) perspective used there extends easily and naturally when larger systems are encountered, as in the treatment in section 9.6 of the method of lines for the heat equation.
Second-order (ordinary) boundary-value problems are solved analytically in section 9.2, Boundary-value Problems: Analytic Tools. The heat equation is solved analytically using separation of variables and eigenfunction expansions in section 9.5.
Laplace transforms are the subject of chapter 10. They are motivated by the notion of sampling a solution function in search of rates of exponential growth or decay (or complex rates signifying oscillations). Although there are ample exercises in the usual manipulations, the spirit of the analysis goes beyond mere manipulation to reach two end points. One is accommodating piecewise continuous and impulsive forcing terms as in section 11.4, Ramps and Jumps, and 11.5, The Unit Impulse Function. The other goal is the sort of qualitative analysis exemplified by such exercises as 15-28 of the chapter 10 chapter exercises, p. 562-563.
Chapter 11 collects a number of familiar non-constant-coefficient solution methods for second-order equations. These can be covered in succession as ordered in the text or sampled from points earlier in the text as your preferences dictate.
For example, sections 11.1, Reduction of Order, and 11.2, Cauchy-Euler Equations, provide a general framework for solving homogeneous diffusion problems in a circle, of which there are examples in section 9.1, Diffusion Models. So 11.1 and 11.2 could immediately follow 9.1 if desired.
Moreover, solving nonhomogeneous diffusion equations in a circle would require section 11.3, Variation of Parameters: Second-order Equations. Hence, that section could join 11.1 and 11.2, immediately following 9.1.
In a completely different way, the idea of specializing from first-order systems back to a second-order equation could motivate a brief study of 11.3, Variation of Parameters: Second-order Equations, immediately following 8.4, Nonhomogeneous Systems: Variation of Parameters.
Similar connections link power series methods and the eigenfunction problems that arise from applying the ideas of section 9.5, Fourier Methods, to the heat equation in a circle. Section 11.4, Power Series Methods, and section 11.5, Regular Singular Points, could be scheduled just after 9.5 to find those eigenfunctions and to characterize their behavior near the origin.
To nurture mathematical maturity, some familiar analytic methods are left to students to develop. Among them are: