Paper 2, Section I, 2A2 A

Differential Equations | Part IA, 2011

(a) For a differential equation of the form dy dx=f(y)\frac{\mathrm{d} y}{\mathrm{~d} x}=f(y), explain how f(y)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

dy dx=y3y22y\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)(x, y)-plane.

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