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# LINEAR ANALYSIS MATH 309

UW

GPA 3.76

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This 2 page Study Guide was uploaded by Addison Beer on Wednesday September 9, 2015. The Study Guide belongs to MATH 309 at University of Washington taught by Margarita Solomyak in Fall. Since its upload, it has received 24 views. For similar materials see /class/192061/math-309-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Math 309 QUIZ PREP SELECTED ANSWERS Winter 2009 I THEORETICAL QUESTIONS i Give an example of a system of two rst order non linear di erential equations 7 2 1 7 1 x2 7 2 2 7 1 2x2 ii Give an example of a non homogeneous system of two rst order linear differential equa tions with non constant coef cients z tx12x21 1 t 2 2 ez1 2 11 PRACTICE PROBLEMS 1 For each of the following matrices a nd all the eigenvalues and eigenvectors b determine if the matrix is diagonalizable and if yes write down an eigenbasis 11 174 1111 01 The rst matrix has eigenvalues 271 and 1 it is not diagonalizable 7quot 2 has algebraic multiplicity two but geometric multiplicity one The second matrix has eigenvalue 1 of algebraic multiplicity three and geometric multiplicity one so it is not diagonalizable either The third matrix has eigenvalues 1 i 2239 it is diagonalizable t 2 Consider the vector functions x1t f and x2t t a Compute the Wronskian Wx1x2 Based on this can you decide if Km and x2 are solutions of a system x Ptx where Pt is a matrix function continuous on 02 Give reasons for your answer 2 71 0 0 71 1 0 0 2 0 0 0 0 1 0 1 b Find the matrix function Pt such that Xlt1gt and x2 are solutions of the system x Ptx Does your answer con rm the conclusion of part a Answers a Wt ett 7 1 which vanishes at t 1 Therefore the coef cient matrix Pt cannot be de ned or continuous at t 1 0 PW 0 7t71 l tt71 1 39 3 Solve the initial value problem a 2x1 2 z10 2 2 1 22 z20 3 z1t 72 gee z2t i gee 4 a Find the general real solution for the system x i 2 5 x T 71 74 b Determine the type of the critical point 00 and stability and sketch a few trajectories c Consider the solution of the system in a satisfying x0 26f What are the quadrants the trajectory passes through In which quadrant will the trajectory be when t 1000 d Consider the solution of the system in a satisfying x0 What are the 7201 2 quadrants the trajectory passes through In which quadrant will the trajectory be when t 1000 Answers a xtcl 5 et02 1Je 3t 71 71 b This is a saddle unstable c The trajectory passes through the quadrants ll I IV It will be in the lV th quadrant at t 1000 d The trajectory always stays in the quadrant ll The answers to e and d illustrate instability The initial conditions were changed very little but eventual behavior is completely di erent 5 a Find the general real solution for the system 2754 271 b What is the type and stability of the point 0 0 A c Find the solution of the system in a satisfying KO 3 d Sketch a few trajectories Answers a xtcl1et62ie3t b This is an unstable node c Take 01 02 1 in a

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