Chapter IV. Theory of Ordinary Differential Equations and Systems

Abstract

This chapter treats the theory of ordinary differential equations, both linear and nonlinear.

Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vector-valued $F$ satisfying a “Lipschitz condition,” that the first-order system $y'=F(t,y)$ has a unique local solution satisfying an initial condition $y(t_0)=y_0$. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for $n^{th}$-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in $F$. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves.

Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to $n^{th}$-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study—Legendre polynomials and Bessel functions.