# sec2.1 - Math 415A Direction Fields to Some Problems from...

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Math 415A: Direction Fields to Some Problems from Section 2.1 The direction ﬁelds are produced using Dﬁeld, a very nice java applet for visualizing direction ﬁelds. The applet can be found on http://math.rice.edu/ dﬁeld/dfpp.html through the courtesy of Prof. John C. Polking from Rice University. Problem 2.1.1. y y´ = -3*y+t+e^(-2*t) -5 -4 -3 -2 -1 0 1 2 3 4 5 t 0 1 2 3 4 5 6 7 8 9 10 Figure 1: As t → ∞ , y ( t ) approaches the line t 3 - 1 9 asymptotically regardless of the initial value y (0) . 1

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Problem 2.1.2. y y´ = 2*y+t²*e^(2*t) -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Figure 2: As t → ∞ , y ( t ) approaches + regardless of the initial value y (0) . 2
Problem 2.1.3. y y´ = -y+t*e^(-t)+1 -4 -3 -2 -1 0 1 2 3 4 t 0 1 2 3 4 5 6 7 8 9 10 Figure 3: As t → ∞ , y ( t ) approaches the the horizontal line y = 1 regardless of the initial value y (0) . 3

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Problem 2.1.4. y y´ = -y/t+3*cos(2*t) -6 -5 -4 -3 -2 -1 0 1 2 t 0 2 4 6 8 10 12 14 16 18 20 Figure 4: As t → ∞ , y ( t ) approaches the function 3 sin 2 t 2 asymptotically regardless of its initial value. In particular, it exhibits oscillating behavior eventually. 4
Problem 2.1.5. y y´ = 2*y+3*e^t -6 -5.5

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sec2.1 - Math 415A Direction Fields to Some Problems from...

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