How do I determine equilibrium solutions and stability for a non-linear ODE?

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Hey Guys,

I am really confused about the first problem on my first problem set in Diff Eq (not auspicious is it? Oh well...)

Draw the Direction field y' = y -y^2. Identify Isoclines and any equlibrium solutions.

I don't understand how to approach this problem because doesn't the y^2 term make the equation non-linear? Also, my professor mentioned that equilibrium solutions could be stable or unstable without telling us what that entailed so can you guys please enlighten me with that , also?

Thank you so much in advance.
 
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You should be able to read your book to find that to figure out eq. points, you set your y' = 0 and solve. Then there's a variety of ways to figure out stability, you can plot your f(y) = y - y^2 and then obviously where it crosses the horizontal axis, those are your fixed points. When your graph is above the horiz. axis, draw an arrow point right "-->" when it's below, draw an arrow point left "<---". Now if both your arrows point towards your fixed point then it's stable, otherwise unstable. You can also find f'(y) and plug in your fixed points.
 

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