Differential Equations Final Exam Topics: Math 308, Exams of Differential Equations

Download Differential Equations Final Exam Topics: Math 308 and more Exams Differential Equations in PDF only on Docsity! Math 308 - Differential Equation Final Exam Topics 1. The basics • What is a differential equation (DE)? • What is a solution to a differential equation? • What is an initial value problem (IVP)? • Classifying differential equations: ordinary or partial, order, linear or nonlinear, autonomous or nonautonomous. 2. First order differential equations • Slope field • Analytical techniques – Separable equations: identify and solve. – Linear equations: identify and solve (integrating factor). • Qualitative techniques for autonomous first order equations y′ = f(y) – Phase line (including interpretation of the graph of f(y) vs. y) – Critical points (i.e. equilibrium solutions) – Stability of equilibria: asymptotically stable, unstable, semistable (Problem 7, p. 84) • Theory: existence, uniqueness • Modeling and applications – Convert a verbal description of a process into a differential equation. – Applications: exponential population growth, drug clearance, mixing, Newton’s law of motion and the velocity of an object, Newton’s law of cooling, logistic equation. • Euler’s Method for numerical approximation (also applies to systems) 3. Linear second order differential equations • Homogeneous or nonhomogeneous, constant coefficients or variable (i.e. “time”-dependent) coef- ficients • Theory: principle of superposition, fundamental set of solutions, linear independence, Wronskian • Find the general solution to ay′′ + by′ + cy = 0 in all cases; solve initial value problems. • Nonhomogeneous: the method of undetermined coefficients • Mechanical vibrations – Mass-spring system: mass, damping or friction coefficient, spring constant. – Free vibrations (i.e. no external forcing), damped or undamped; natural frequency, “quasi- frequency” – Forced vibrations (external forcing, beats, resonance, transient response, steady state re- sponse, amplitude, phase shift) 4. Systems of first order equations (topics applicable to linear or nonlinear systems) • What is a system of first order differential equations? • What is a solution to a system of first order differential equations? • Convert a second order differential equation into a first order system. 1