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# LINEAR ANALYSIS MATH 309

UW

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This 2 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 309 at University of Washington taught by Margarita Solomyak in Fall. Since its upload, it has received 21 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 Preparation for the QUIZ Winter 2009 The quiz will be on Friday7 January 23 It will take all 50 minutes and will cover the Sections 71775 I decided to exclude 76 from this quiz You can bring one standard 8 gtlt 11 sheet of handwritten notes no printed materials7 2 sided is OK No calculators or other electronic devices are allowed The problems below should give you an idea of the types of problems that will be on the quiz Obviously7 it will not be as long this just gives a range of possibilities I THEORETICAL QUESTIONS i Give an example of a system of two rst order non linear di erential equations ii Give an example of a non homogeneous system of two rst order linear differential equa tions with non constant coef cients iii Give the de nition of the following a diagonalizable matrix b nonsingular matrix II PRACTICE PROBLEMS 1 For each of the following matrices7 a nd all the eigenvalues and eigenvectors b determine if the matrix is diagonalizable7 and if yes7 write down an eigenbasis 0 0 11 174 1 1111 1 01 t 2 Consider the vector functions x1t f and x2t t a Compute the Wronskian Wx1x2 Based on this7 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 1 0 0 0 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 3 Solve the initial value problem 4 a Find the general real solution for the system 2 5 71 74 X X b Determine the type of the critical point 07 0 and stability7 and sketch a few trajectories 72 c Consider the solution of the system in a satisfying x0 2 01 What are the quadrants the trajectory passes through In which quadrant will the trajectory be when t 1000 Hint you don t need a formula for the solution to answer this question The quadrants are numbered as follows U I U V i i i i i 7201 d Consider the solution of the system in a satisfying x0 2 What are the quadrants the trajectory passes through In which quadrant will the trajectory be when t 1000 5 a Find the general real solution for the system a u 574 271 b What is the type and stability of the point 07 0 c Find the solution of the system in a satisfying KO 3 d Sketch a few trajectories

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