To show that if the initial conditions satisfy x 0 + y 0 = 1 , then x ( t ) + y ( t ) = 1 for all t , for the system x ˙ = ax c - ϕ x, y ˙ = by c - ϕ y where ϕ = ax c + by c . To show that all the trajectories starting in the positive quadrant are attracted to the invariant line x + y = 1 . To draw the phase portraits in the ( x, y ) plane for c = 1, c > 1, and c < 1 . Concept Introduction: It can be shown that x ( t ) + y ( t ) = 1 for all t , by adding the given equations. The Jacobian matrix at a general point ( x, y ) is given by J = ( ∂ x ˙ ∂ x ∂ x ˙ ∂ y ∂ y ˙ ∂ x ∂ y ˙ ∂ y )

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Chapter 6.4, Problem 8E

Interpretation Introduction

Interpretation:

To show that if the initial conditions satisfy x0+ y0 = 1, then x(t) + y(t) = 1 for all t, for the system x˙ = axc- ϕx, y˙ = byc- ϕy where ϕ = axc+ byc. To show that all the trajectories starting in the positive quadrant are attracted to the invariant line x + y = 1. To draw the phase portraits in the (x, y) plane for c = 1, c > 1, and c < 1.

Concept Introduction:

It can be shown that x(t) + y(t) = 1 for all t, by adding the given equations.

The Jacobian matrix at a general point (x, y) is given by

J = (∂x˙∂x∂x˙∂y∂y˙∂x∂y˙∂y)

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