Consider the system dx dt = x3 dy dt = y + y2. It has

Chapter , Problem 17

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Consider the system dx dt = x3 dy dt = y + y2. It has equilibrium points at (0, 0) and (0, 1). (a) Find the linearized system at (0, 0). (b) Find the eigenvalues and eigenvectors and sketch the phase portrait of the linearized system at (0, 0). (c) Find the linearized system at (0, 1). (d) Find the eigenvalues and eigenvectors and sketch the phase portrait of the linearized system at (0, 1). (e) Sketch the phase portrait of the nonlinear system. [Hint: The system decouples, so first draw a phase line for each of the individual equations.] (f) Why do the phase portraits for the linearized systems and the phase portrait for the nonlinear system near the equilibrium points look so different?

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