Download Math 287 Exam Review: Differential Equations and Linear Operators - Prof. Robert A. Schwen and more Exams Linear Algebra in PDF only on Docsity!
Math 287 Exam 1, sections 1.1 – 1.5, 2.1 – 2.
Vocabulary (Chapter 1) o Mathematical Model o Dynamical system o Ordinary Differential Equation o Initial Value Problem o Malthus population model o General solution to a DE o A particular solution to a DE o Equilibrium solution o Stable o Unstable o Semistable
o Isocline o Basin of attraction
o The order of a differential equation o Separable differential equation o Implicitly defined function o Explicitly defined function o Descretization error o Roundoff error o Big-oh notation (order of the error) o Richardson’s extrapolation o Existance theorem o Uniqueness theorem o Picard’s Existance and Uniqueness theorem
(Chapter 2) o Linear Operator o Linear Equation o Homogeneous linear equation o Nonhomogeneous linear equation
o Linear differential equation o Steady state solution o Transient Solution o
Computations o Differentiate o Implicit differentiation o Integration using substitution, integration by parts and partial fractions o Use Euler’s Method, Runga-Kutta order 4, Richardson’s extrapolation o Transform an implicit solution to a differential equation into an explicit solution o Given the general solution to a differential equation solve an IVP o Calculate the integrating factor for a first order linear differential equation
Proofs o The nonhomogeneous superposition principle for first order linear differential equations
Determinations and Demonstrations (chapter 1) o That a given function is or is not a solution to a differential equation or IVP o That a given IVP does or does not satisfy Picard’s existence uniqueness theorem o Draw the isoclines and direction field of a first order differential equation o Draw a solution to an IVP through a direction field o Solve a differential equation using separation of variables o Transform an Euler-homogeneous differential equation into a separable differential equation, given the substitution y=vt o Match a differential equation with its direction field o Draw the direction field of a differential equation o Analyze a direction field (checklist in section 1.2) o Find the equilibriums of an autonomous differential equation and the basins of attraction for each equilibrium
(chapter 2) o Classify differential equations by order, linearity, as homogeneous or nonhomogeneous o Use the homogeneous superposition principle, the nonhomogeneous superposition principle, and the nonhomogeneous principle to identify solutions of a linear equation and to build the solution set of a linear equation o Solve nonhomogeneous linear differential equations using variation of parameters and integrating factors o Solve a nonhomogeneous linear initial value problem o Set up the IVPs to model application problems involving growth, decay, mixtures, heating, cooling o Solve IVPs modeling application problems involving growth, decay, mixtures, heating, cooling
