Linear Ordinary Differential Equations, a text for advanced undergraduate or beginning graduate students, presents a thorough development of the main topics in linear differential equations. A rich collection of applications, examples, and exercises illustrates each topic. The authors reinforce students' understanding of calculus, linear algebra, and analysis while introducing the many applications of differential equations in science and engineering. Three recurrent themes run through the book: The methods of linear algebra are applied directly to the analysis of systems with constant or periodic coefficients and serve as a guide in the study of eigenvalues and eigenfunction expansions. The use of power series, beginning with the matrix exponential function, leads to the special functions solving classical equations. Techniques from real analysis illuminate the development of series solutions, existence theorems for initial value problems, the asymptotic behavior solutions, and the convergence of eigenfunction expansions.
Preface Chapter 1: Simple Applications. Introduction Compartment systems Springs and masses Electric circuits Notes Exercises Chapter 2: Properties of Linear Systems. Introduction Basic linear algebra First-order systems Higher-order equations Notes Exercises Chapter 3: Constant Coefficients. Introduction Properties of the exponential of a matrix Nonhomogeneous systems Structure of the solution space The Jordan canonical form of a matrix The behavior of solutions for large t Higher-order equations Exercises Chapter 4: Periodic Coefficients. Introduction Floquet's theorem The logarithm of an invertible matrix Multipliers The behavior of solutions for large t First-order nonhomogeneous systems Second-order homogeneous equations Second-order nonhomogeneous equations Notes Exercises Chapter 5: Analytic Coefficients. Introduction Convergence Analytic functions First-order linear analytic systems Equations of order n The Legendre equation and its solutions Notes Exercises Chapter 6: Singular Points. Introduction Systems of equations with singular points Single equations with singular points Infinity as a singular point Notes Exercises Chapter 7: Existence and Uniqueness. Introduction Convergence of successive approximations Continuity of solutions More general linear equations Estimates for second-order equations Notes Exercises Chapter 8: Eigenvalue Problems. Introduction Inner products Boundary conditions and operators Eigenvalues Nonhomogeneous boundary value problems Notes Exercises Chapter 9: Eigenfunction Expansions. Introduction Selfadjoint integral operators Eigenvalues for Green's operator Convergence of eigenfunction expansions Extensions of the expansion results Notes Exercises Chapter 10: Control of Linear Systems. Introduction Convex sets Control of general linear systems Constant coefficient equations Time-optimal control Notes Exercises Bibliography.
