Science:Math Exam Resources/Courses/MATH215/December 2013/Question 07 (a)
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Question 07 (a) 

Consider the following initial value problem for a firstorder autonomous ODE: (a) Find the equilibrium solutions (critical points), determine the stability (stable or unstable) of each, and find for all possible values of the initial condition . 
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 

The equilibrium solutions are the constant solutions where . To assess the stability, consider what the sign of the derivative tells you about solutions near the equilibrium points. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Setting , we have so . Looking at the sign of , we have . Thus, if we start below , increases; if we start above but below , decreases. Either way, we approach so is a stable equilibrium solution. If we start below but above , decreases and moves away from () and if we start above , decreases and approaches (). However, because the solution moves away from locally on at least one side, is an unstable equilibrium solution. All possible values for include 0 and 1. If , as . If , as . Finally, if then as . 