Paper 2, Section I, $2 A$

(a) For a differential equation of the form $\frac{\mathrm{d} y}{\mathrm{~d} x}=f(y)$, explain how $f^{\prime}(y)$ can be used to determine the stability of any equilibrium solutions and justify your answer.

(b) Find the equilibrium solutions of the differential equation

$\frac{\mathrm{d} y}{\mathrm{~d} x}=y^{3}-y^{2}-2 y$

and determine their stability. Sketch representative solution curves in the $(x, y)$-plane.

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