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# LINEAR ANALYSIS MATH 309

UW
GPA 3.76

Margarita Solomyak

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COURSE
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Margarita Solomyak
TYPE
Class Notes
PAGES
1
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 1 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 19 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 Winter 2009 GRAPHICAL REPRESENTATION OF SOLUTIONS FOR SYSTEMS OF TWO LINEAR ODE WITH CONSTANT COEFFICIENTS We consider the system where x xAx 1 and A is a constant 2 gtlt 2 matrix Phase plane the 1 2 plane Trajectory solution curve this is a parametric curve given by 1 x1t 2 z2t foo lt t lt 00 By Theorem 712 Existence and Uniqueness of Solutions for systems of linear ODE if you pick any point in the phase plane say 3 2 and any time to say to 0 then there is one and only one trajectory passing through the given point at the time to Direction eld plot of velocity vectors for solution curves The velocity vector for the parametric motion x1tx2t is x t represented by the vector X Due to the equation I we have that x AX Thus for example the 3 The direction eld leads to a graphical method for nding approximate solutions velocity vector at the point 32 is given by A Equilibrium solution solution of the form z1t cl z2t 02 The corresponding trajectory is a single point Equilibrium points are found by solving Ax 0 since the velocity in equilibrium is zero Phase portrait plot showing some representative trajectories Why and how do we plot by hand This gives a feeling for the qualitative features of the solution Computer generated plots may be misleading see eg Figure 754 what is wrong with it We are NOT trying to make perfect plots just present some essential features of the solutions sometimes even exaggerating them

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