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# Ordinary Differential Equation MATH 302

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This 2 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 302 at College of William and Mary taught by Junping Shi in Fall. Since its upload, it has received 8 views. For similar materials see /class/231142/math-302-college-of-william-and-mary in Mathematics (M) at College of William and Mary.

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Date Created: 10/29/15

Recurrent Sequences difference equations and differential equations Quick review of Math 302 Differential Equations 1 y Ay By 240 a0 20 a1 for which the solution is 2 yt 015 626A2t7 where 1 and 2 are the roots of quadratic characteristic equation A2 A B If 1 A2 then the solution is 3 yt 615 cgteAlt Recurrence relations A recurrent sequence is de ned as 4 MM f90n71727 H 7nk717 900 107 901 117 H 7k71 kil 4 is also called a dz erence equation The number k is the order of the equation or relation A k th order linear recurrent sequence is generated by a linear equation 5 nk bng n bn1n1 bn290n2 bnk71nk717 900 0107951 0117 7k71 akil The most common ones are rst order recurrent sequence 6 95n1 100179507950 107 or second order recurrent sequence 7 zn2 fnmmzn1m0 a07m1 al An autonomous rst order recurrent sequence 8 95n1 9607 900 0107 is also often called a map Linear recurrence relations The theory of linear recurrent sequence is very similar to that of linear ordinary differ ential equation For example7 a second order linear recurrent sequence 9 n2 14 Bn17 900 0107901 a1 The solution is given by 10 25 cw mg where 1 and 2 are the roots of quadratic characteristic equation A2 A B and 01 02 are to be determined by the initial conditions If 1 A2 then the solution is 11 n 61A 6271A For non homogeneous linear recurrent sequence 12 n2 AM Bn1 07900 107 951 117 Notice that it is possible to make a change of variable yn zn k so that yn satis es 9 and k is determined by A B and C Nonlinear recurrence relations Since linear autonomous recurrent equations always have solution formulas most prob lems in mathematics competitions are either non autonomous or nonlinear However the methods for linear equation are still very useful and sometimes non autonomous or nonlin ear maybe reduced to linear autonomous equation via certain smart change of variables In general there is no explicit solution formula for non autonomous or nonlinear equation even for a simple equation like mn Amn1 7 mn Logistic equation Indeed the solutions of logistic equation are chaotic when the parameter A is large If you have the textbook of Math 302 Blanchard Devaney Hall Differential Equations Chapter 8 of that book has a good introduction for logistic equation For the 1st order nonlinear autonomous recurrent equation my fmn a xed point z is the one satisfying z Note that for such equation mn f m0 where f y ff 1y So it is also often called iterated sequence A xed point z is attracting if f y a z for all y near m and it is repelling if f y goes away from x for all y near z There are also xed point neither attracting nor repelling A xed point z is attracting if lf ml lt 1 and it is repelling if lf ml gt 1 The iterated sequence can be drawn in z 7 y coordinate system with so called web diagram Finally one can have a system of difference equations 13 n1 Am Bym yn1 0 Dyn and the solution is given in a form mmyn 0102 1 0304 3 and 1 2 are the eigenvalues of the matrix lt3 5 Some weblinks on difference equations

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