DE.05.4.Literacy - y 12 Differential...

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Mathematica Authors: Bruce Carpenter, Bill Davis and Jerry Uhl ©2001-2007 Publisher: Math Everywhere, Inc. Version 6.0 DE.05 First Order Differential Equations LITERACY · L.1) You are given this diffeq to analyze: y £ @ t D ã 0.2 H - 3 + y @ t DL H - 1 + y @ t DL y @ 0 D ã starter Then you look at: f @ t, y D 0.2 H - 3 + y L H - 1 + y L And you say to yourself, "One single phase line is enough." - 1 1 2 3 4 5 6 t 1 2 3 4 y Look again at the formula for f @ t, y D : f @ t, y D 0.2 H - 3 + y L H - 1 + y L What is it about the formula for f @ t, y D that made you know you could get by with just one phase line? What information does the phase line give? · L.2) Here's an autonomous diffeq: y £ @ t D ã H 2 - y @ t DL H - 7 + y @ t DL H - 4 + y @ t DL y @ 0 D ã starter And the phase line for it: - 1 1 2 3 4 5 t 2 4 6 8 y The phase line indicates four different types of solutions. Pencil in a sample of each. For what starting values do you expect constant solutions? Which dot on the phase line signals the place where you expect extreme sensitivity to errors in starter data on y @ 0 D ? · L.3) Here's a new autonomous diffeq containing some partially random coefficients: y £ @ t D ã 1.10733 H 1 - 0.37826 y @ t DL H 1 - 0.166008 y @ t DL H 1 - 0.109701 y @ t DL y @ 0 D ã starter Look at this plot: - 1.0 - 0.5 0.5 1.0 f @ t,y D 2 4 6 8 10 12 y Points on the curve are of the form: 8 f @ t, y D , y < 8 1.10733 H 1 - 0.37826 y L H 1 - 0.166008 y L H 1 - 0.109701 y L , y < But the real issues here are: Ø How are the arrowheads on the phase line related to the plot of 8 f @ t, y D , y < ? Ø How are the dots on the phase line related to the plot of 8 f @ t, y D , y < ? Ø Remembering that y £ = f @ t, y D , explain why it had to turn out this way. · L.4) You are given this diffeq to analyze: y £ @ t D ã - 2 - Sin @ t D + y @ t D y @ 0 D ã starter The first thing you do is look at: f @ t, y D - 2 + y - Sin @ t D And you say to yourself: "One single phase line is not enough. If I want to use phase lines,
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DE.05.4.Literacy - y 12 Differential...

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