A First Course in Differential Equations

• Chapter 1. First-order equations. Separable, linear, and autonomous equations; equilibrium solutions, stability and bifurcation. Other special types of equations, for example, Bernoulli, exact, and homogeneous equations, are covered in the Exercises with generous guidance. Many applications are discussed from science, engineering, economics, and biology.
• Chapter 2. Second-order linear equations. The emphasis is on equations with constant coefficients, both homogeneous and nonhomogeneous, with most examples being spring-mass oscillators and electrical circuits. Other than Cauchy–Euler equations, variable coefficient equations are not examined in detail. There are three optional sections covering reduction of order, higher-order equations, and steady-state heat transfer, which deals with simple boundary value problems.
• Chapter 3. Laplace transforms. The treatment is standard, but without overemphasizing partial fraction decompositions for inversion. Use of the enclosed table of transforms is encouraged. This chapter can be covered at any time after Chapter 2.
• Chapter 4. Linear systems. This chapter deals only with two-dimensional, or planar, systems. It begins with a discussion of equivalence of linear systems and second-order equations. Linear algebra is kept at a minimum level, with a very short introductory section on notation using vectors and matrices. General solutions are derived using eigenvalues and eigenvectors, and there are applications to chemical reactors (compartmental analysis), circuits, and other topics. There is a thorough introduction to phase plane analysis and simple geometric methods.
• Chapter 5. Nonlinear systems. This chapter revolves around applications, e.g., classical dynamics, circuits, epidemics, population ecology, chemical kinetics, malaria, and more. Typically, inclusion of this chapter requires a 4-credit semester course.
• Chapter 6. Computation of solutions. This brief chapter first discusses the Picard iteration method, and then numerical methods. The latter include the Euler and modified Euler methods, and the Runge–Kutta method. All or parts of this chapter can be covered or referred to at any time during the course.

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