After a while it started to sink in that these texts I intended to supplement had bloated out over the years and turned into 800 page monsters. So my goal shifted from supplementing these texts to replacing them. The manuscript is shaping up at about 350 pages.
Chapter 1 treats single differential equations, linear and nonlinear, with emphasis on first and second order equations. The first section provides a self contained development of exponential functions eat, as solutions of the differential equation dx/dt=ax. We allow a to be complex, and also provide a self contained treatment of the trigonometric functions. The following two sections treat first order equations, and then we quickly move to second order equations. We emphasize equations arising from Newton's laws, applied to 1D motion, first without friction and then with friction. We are motivated to study linearizations of these equations, which occupy the rest of the chapter.
Chapter 2 is devoted to linear algebra. This includes definitions of vector spaces and linear transformations, the notion of basis and dimension of a vector space, and representation of a linear transformation by a matrix, in terms of a choice of basis. We have a treatment of determinants of square matrices, followed by a discussion of eigenvalues and eigenvectors of a linear transformation, and then of generalized eigenspaces. We also discuss several special classes of linear transformations, particularly nilpotent transformations, and also self-adjoint, skew-adjoint, unitary, and orthogonal transformations.
In Chapter 3 we apply the material of Chapter 2 to the study of linear systems of differential equations. The first section is devoted to the matrix exponential, extending the results that began Chapter 1. Section 2 provides a complementary approach to trigonometric functions, and extends the scope to the sine and cosine of matrices. We proceed to various topics on linear systems, first constant coefficient and homogeneous, then non-homogeneous, then variable coefficient. An integral formula, Duhamel's formula, is seen to provide an elegant replacement for the method of variation of parameters. Several sections are devoted to applications to electrical circuits, spring systems, and the Frenet-Serret equations for space curves. We end the chapter with a treatment of power series methods for systems with analytic coefficients and a treatment of systems with regular singular points, and show how this material applies to single equations of second order.
Chapter 4 treats nonlinear systems of differential equations. We prove results on existence and uniqueness of solutions, and dependence on initial conditions and other parameters. We discuss geometrical aspects, interpreting solving the initial value problem to constructing the flow generated by a vector field. We bring in the phase portrait and discuss the nature of critical points of a vector field. We resume the discussion from Chapter 1 of equations arising from applying Newton's laws, this time to the interaction of k bodies in n-dimensional space. We produce Newton's solution to the planetary motion problem. We discuss variational problems, and show how they allow another attack on problems in Newtonian physics. We also discuss some nonlinear systems that arise in mathematical biology, such as predator-prey equations and competing species equations. We end with a discussion of how, in dimension 3 and higher, flows can have chaotic behavior.