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. 1. Simple Applications. Introduction; Compartment systems; Springs and masses; Electric circuits; Notes; Exercises; 2. Properties of Linear Systems. Introduction; Basic linear algebra; First-order systems; Higher-order equations; Notes; Exercises; 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; 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; 5. Analytic Coefficients. Introduction; Convergence; Analytic functions; First-order linear analytic systems; Equations of order n; The Legendre equation and its solutions; Notes; Exercises; 6. Singular Points. Introduction; Systems of equations with singular points; Single equations with singular points; Infinity as a singular point; Notes; Exercises; 7. Existence and Uniqueness. Introduction; Convergence of successive approximations; Continuity of solutions; More general linear equations; Estimates for second-order equations; Notes; Exercises; 8. Eigenvalue Problems. Introduction; Inner products; Boundary conditions and operators; Eigenvalues; Nonhomogeneous boundary value problems; Notes; Exercises; 9. Eigenfunction Expansions. Introduction; Selfadjoint integral operators; Eigenvalues for Green's operator; Convergence of eigenfunction expansions; Extensions of the expansion results; Notes; Exercises; 10. Control of Linear Systems. Introduction; Convex sets; Control of general linear systems; Constant coefficient equations; Time-optimal control; Notes; Exercises; Bibliography.
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