Sometimes a constant equilibrium solution has the property that solutions lying on one side of the...

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Advance Math

Sometimes a constant equilibrium solution has the property thatsolutions lying on one side of the equilibrium solution tend toapproach it, whereas solutions lying on the other side depart fromit. In this case the equilibrium solution is said to be semistable.Consider the equation dy/dt = y^2(4 âˆ’ y 2 ) = f(y), where y(0) = y0and âˆ’âˆž < y0 < âˆž.
(i) Sketch the graph of f(y) versus y.
(ii) Determine the critical points. (iii) Classify each one asasymptotically stable, unstable, or semistable. (iv) Illustrateseveral solutions in the ty-plane that illustrate how the differentsolutions depend upon y0.

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a If y is an equilibrium solution then dydt 0 identically so 0 dy dt k1 y 2 hence yt 1 identically b 1 2 k y fy dy dt Since dydt k1 y 2 0 for y 6 1 then y is increasing for y 1 and y 1 c Separating the See Answer

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