Science:Math Exam Resources/Courses/MATH102/December 2014/Question A 04

MATH102 December 2014

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Question A 04
Which differential equation below is the appropriate one for the slope field shown in the figure? Slope Field $y'=y(y-1)$ $y'=y(1-y)$ $y'=y(1-y)^{2}$ $y'=y^{2}(1-y)$ $y'=y^{2}$

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
How to relate direction field to differential equation?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. First we observe there are two steady states (where y’=0) $y=0$ and $y=1$. Then from the direction of those vectors, we find $y=0$ is unstable and $y=1$ is stable for $y<1$, unstable for $y>1$. To obtain the correct stability of those steady states, we must have ${\begin{aligned}y'<0,\quad &{\text{when }}y<0,\\y'>0,\quad &{\text{when }}0<y<1,\\y'>0,\quad &{\text{when }}1<y,\end{aligned}}$ The first observation eliminates options a) and d). The third observation rules out option b). Option e) is not valid because a steady state must exist when $y=1$. Thus the only possible answer is (c).

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