Differential Equations, Vibrations, and Controls I

(3 Credit Hours) · Undergraduate

First of a two-course sequence that introduces methods of differential-equation solution together with common engineering applications in vibration analysis and controls. Includes second-order, linear differential equations, feedback control, single-degree of freedom vibrations, numerical solutions to systems of ordinary differential equations, and partial differential equations. Fall. (SI) The study of ordinary differential equations (ODEs) has been a centerpiece in both pure and applied mathematics, such as in mathematical analysis, dynamical systems and mathematical modeling. The aim of this course is to give a thorough treatment on the fundamental theory of ODEs and the methods of solving ODEs. Major topics: Review of first order equations, Basic theory of linear differential equations, Variation of parameters, Principle of superposition, Wronskian, Abel's formula, Adjoint and self-adjoint equations, Lagrange and Green's identities, Sturm's separation and comparison theorems, Linear differential systems, Series solutions of second order linear differential equations, Method of Frobenius, Initial value problems, Lipschitz condition, Picard's method of successive approximations, Existence and uniqueness of solution, Gronwall's inequality, Continuous dependence on initial value. (SC) This course aims to teach students methods and applications of first and second order differential equations, as well as providing problem solving strategies and qualitative analysis, to different engineering applications.