Science:Math Exam Resources/Courses/MATH100 B/December 2023/Question 20
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Question 20 

Sketch the phase line for the differential equation Clearly indicate the steady states (also called equilibrium points), and indicate with arrows where is increasing or decreasing on either side of each steady state. 
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 

Recall: if is at an equilibrium point, then the function is not changing with time. Given that the derivative is the rate of change, what is happening to the derivative at an equilibrium point? 
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. We know that if the function is at an equilibrium point, then it is not changing, which means the derivative of the function is zero. Since we are given the differential equation in factored form, we can read off the zeros easily: when ; these are the equilibrium points. To determine if the function is increasing or decreasing on either side of the equilibrium points, we look at whether is positive or negative. When , is positive. When , is negative. When , is positive. And when , is negative. So, we have that the function is increasing for , decreasing for , increasing for , and decreasing for , which means is an unstable equilibrium, is a stable equilibrium, and is an unstable equilibrium. The phase line representing all of this is given below. 