Science:Math Exam Resources/Courses/MATH215/April 2014/Question 01 (c)
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Question 01 (c) 

Consider . For what possible initial conditions does ? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

You do not need to solve the equation in order to answer this question. 
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Solution  

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Please rate my easiness! It's quick and helps everyone guide their studies. The key is to realize that the question is asking for qualitative information which can be inferred by sketching trajectories for the ODE. In particular, it is not necessary to solve this ODE, which would require much more work. We first observe that the ODE has equilibrium (fixed) points at y = 0, y = 1, and y = 2. This means that any trajectories starting at 0, 1, or 2, will remain at that constant value for all time. In particular, we see that the initial condition y(0) = 1 satisfies . To sketch trajectories, we record the following signs of in the four regions separated by the fixed points: From this, we draw the following conclusions:
Therefore, the only initial conditions for which are . 