Draw a direction field for the differential equation y' = -y(2-y). Based on the direction field, determine the behavior of y as t→∞. If this behavior depends on the initial value of y at t = 0, describe this dependency. The two equilibrium solutions are y(t) = 0 and y(t) = 2 Solutions with initial values greater than 2 diverge from the solution y(t) = 2. Solutions with initial values between 0 and 2 decrease toward the solution y(t) = 0. ✓ Solutions with initial values less than 0 increase toward the solution y(t) = 0. ✓
Draw a direction field for the differential equation y' = -y(2-y). Based on the direction field, determine the behavior of y as t→∞. If this behavior depends on the initial value of y at t = 0, describe this dependency. The two equilibrium solutions are y(t) = 0 and y(t) = 2 Solutions with initial values greater than 2 diverge from the solution y(t) = 2. Solutions with initial values between 0 and 2 decrease toward the solution y(t) = 0. ✓ Solutions with initial values less than 0 increase toward the solution y(t) = 0. ✓
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Draw a direction field for the differential equation y' = -y(2- y).
Based on the direction field, determine the behavior of y as t →∞.
If this behavior depends on the initial value of y at t = 0, describe
this dependency.
The two equilibrium solutions are
y(t) = 0
and y(t)
= 2
✓
Solutions with initial values greater than 2
diverge from the solution y(t) = 2.
Solutions with initial values between 0 and 2
decrease toward the solution y(t) = 0.
Solutions with initial values less than 0
increase toward the solution y(t) = 0.
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