Lesson30 First Order Difference Equations Slides

Lesson 30First Order Difference EquationsMath 20April 27, 2007AnnouncementsPS 12 due Wednesday, May 2MT III Friday, May 4 in SC Hall AFinal Exam (tentative): Friday, May 25 at 9:15amProblem 1Make cobweb diagrams of lots of different linear differenceequations. Here are a few to get you started:yk+1 = −1/2yk + 1yk+1 = 3/2yk + 1yk+1 = 1/2yk + 1yk+1 = −yk + 1yk+1 = −3/2yk + 1yk+1 = yk + 1Fun with appletshttp://math.bu.edu/DYSYS/applets/linear-web.htmlFactThe difference equation yk+1 = ayk + b has a uniqueequilibrium when a = 1. The equilibrium is stable when |a| < 1.Problem 2Make a conjecture: “The difference equation yk+1 = ayk + b hasa stable equilibrium when a and b satisfy the conditions that. . . ”Problem 2Make a conjecture: “The difference equation yk+1 = ayk + b hasa stable equilibrium when a and b satisfy the conditions that. . . ”FactThe difference equation yk+1 = ayk + b has a uniqueequilibrium when a = 1. The equilibrium is stable when |a| < 1.FactThe general solution to the linear homoegeneous differenceequation is yk+1 = ayk , y0 = y0 isyk = ak y0Problem 3Guess and check solutions to the following differenceequations:(i) yk+1 = 2yk ,(ii) yk+1 = −1/2yk ,(iii) yk+1 = ayk ,y0 = 1y0 = 1/3y0 = y0Problem 3Guess and check solutions to the following differenceequations:(i) yk+1 = 2yk ,(ii) yk+1 = −1/2yk ,(iii) yk+1 = ayk ,y0 = 1y0 = 1/3y0 = y0FactThe general solution to the linear homoegeneous differenceequation is yk+1 = ayk , y0 = y0 isyk = ak y0SolutionWe getyk = (−1/2)k (−2/3) + 2/3Problem 4(a) Find the equilibrium solution to yk+1 = −1/2yk + 1.(b) Find the general solution to yk+1 = −1/2yk , y0 = c.(c) Add the two together and choose c to solve the equationwith initial conditionsyk+1 = −1/2yk + 1,y0 = 0Problem 4(a) Find the equilibrium solution to yk+1 = −1/2yk + 1.(b) Find the general solution to yk+1 = −1/2yk , y0 = c.(c) Add the two together and choose c to solve the equationwith initial conditionsyk+1 = −1/2yk + 1,y0 = 0SolutionWe getyk = (−1/2)k (−2/3) + 2/3This establishes our first conjecture about stability, too.Solving the inhomogenous equationFactThe linear first-order difference equationyk+1 = ayk + bhas solutions given bybbaky0 −+if a = 1yk =1 − a1 − ay0 + kbif a = 1